Pleiotropy, Sex-Dependent Allelic Effects and GE Interactions

Polygenic Variation Maintained by Balancing Selection: Pleiotropy, Sex-Dependent Allelic Effects and G ϫ E Interactions

Michael Turelli*,1 and N. H. Barton† *Section of Evolution and Ecology and Center for Population Biology, University of California, Davis, California 95616 and †Institute of Cell, Animal and Population Biology, University of Edinburgh, Edinburgh EH9 3JT, United Kingdom Manuscript received May 29, 2003 Accepted for publication October 17, 2003

ABSTRACT We investigate three alternative selection-based scenarios proposed to maintain polygenic variation: pleiotropic balancing selection, G ϫ E interactions (with spatial or temporal variation in allelic effects), and sex-dependent allelic effects. Each analysis assumes an additive polygenic trait with n diallelic loci under stabilizing selection. We allow loci to have different effects and consider equilibria at which the population mean departs from the stabilizing-selection optimum. Under weak selection, each model produces essentially identical, approximate allele-frequency dynamics. Variation is maintained under pleiotropic balancing selection only at loci for which the strength of balancing selection exceeds the effective strength of stabilizing selection. In addition, for all models, polymorphism requires that the population mean be close enough to the optimum that directional selection does not overwhelm balancing selection. This balance allows many simultaneously stable equilibria, and we explore their properties numerically. Both spatial and temporal G ϫ E can maintain variation at loci for which the coefﬁcient of variation (across environments) of the effect of a substitution exceeds a critical value greater than one. The critical value depends on the correlation between substitution effects at different loci. For large positive ␳2 Ͼ correlations (e.g., ij 3/4), even extreme ﬂuctuations in allelic effects cannot maintain variation. Surpris- ingly, this constraint on correlations implies that sex-dependent allelic effects cannot maintain polygenic variation. We present numerical results that support our analytical approximations and discuss our results in connection to relevant data and alternative variance-maintaining mechanisms.

T remains a challenge for evolutionary geneticists to lelic effects (treating spatial and temporal heterogene- I understand the additive genetic variance observed ity separately), and sex-dependent allelic effects. The for most traits in most populations. Given the ubiquity thread that unites these scenarios is that, under weak of additive genetic variation, it is natural to seek an selection, each produces very similar allele-frequency explanation in terms of ubiquitous forces. Lande (1975) dynamics and polymorphism conditions. An empirical proposed mutation-selection balance. However, over motivation for these analyses is that alleles of intermedi- the past 25 years, attempts to explain standing levels ate frequency seem to contribute to phenotypic varia- of quantitative genetic variation in terms of mutation- tion in natural populations (e.g., Mackay and Langley selection balance have been at best only partially success- 1990; Long et al. 2000). Such polymorphisms are incom- ful (e.g., Caballero and Keightley 1994; Charles- patible with mutation-selection balance for plausible worth and Hughes 2000; but see Zhang and Hill levels of selection and mutation. 2002). One alternative is that some form of balancing The mathematical motivation for our analyses is selection, unconnected to the trait of interest, may ac- Wright’s (1935) demonstration that stabilizing selec- count for persistent polymorphism at the underlying tion tends to eliminate polygenic variation. Using a weak- loci (e.g., Robertson 1965; Bulmer 1973; Gillespie selection approximation, he showed that at most one 1984; Barton 1990). In contrast to such pleiotropic ex- locus is expected to remain polymorphic at equilibrium. planations, balancing selection might arise from varia- More recent analyses of strong selection (Nagylaki tion in the effects of alleles that contribute to the trait, 1989; Bu¨rger and Gimelfarb 1999) have found that for instance, through genotype-by-environment (G ϫ E) two-locus polymorphisms can be stably maintained with interactions. Here we explore four scenarios in which sufﬁciently strong selection and sufﬁcient interlocus variance-depleting stabilizing selection interacts with pleio- variation in allelic effects. We provide new simulations tropic balancing selection, environment-dependent al- that further illustrate the restrictive conditions needed to maintain even stable two-locus polymorphisms for additive traits under stabilizing selection and loose linkage. 1Corresponding author: Section of Evolution and Ecology, University of California, 1 Shields Ave., Davis, CA 95616. Robertson (1965) proposed that additive variation E-mail: [email protected] may be maintained by pleiotropically induced overdom-

Genetics 166: 1053–1079 ( February 2004) 1054 M. Turelli and N. H. Barton inant selection, which counteracts the effects of stabiliz- that sex-dependent allelic effects do not stably maintain ing selection. His conjecture was explored analytically polygenic variation for additive traits. by Bulmer (1973) for diallelic loci and extended to All of our analyses assume that selection is weak multiple alleles by Gillespie (1984). Both assumed enough relative to recombination that linkage disequi- equal allelic effects across loci, symmetric overdomi- librium is negligible. We also assume diallelic loci. It is nance of equal intensity at all loci, and that the popula- not clear to us how restrictive this assumption is. In tion mean at equilibrium coincided with the optimal models of mutation-selection balance, two-allele and trait value. They found lower bounds on the intensity continuum-of-allele models give similar results provided of overdominant selection required to maintain stable that the alleles responsible for variation are rare (Tur- multilocus polymorphisms. Our diallelic analyses gener- elli 1984; Slatkin and Frank 1990). However, when alize theirs and that of Zhivotovsky and Gavrilets loci are highly polymorphic (as might occur under bal- (1992), by allowing for unequal allelic effects and arbi- ancing selection), continuum-of-allele models can give trary overdominance across loci and by considering the qualitatively different results (Bu¨rger 1999; Waxman simultaneous stability of alternative equilibria at which and Peck 1999; Waxman 2003). Nevertheless, we be- the population mean can depart from the optimum. lieve that models with two alleles are a better approxi- Gillespie and Turelli (1989) showed how balancing mation to reality (where there will usually be a few selection could arise at individual loci by averaging over discrete alleles) than are a continuum of alleles, particu- randomly fluctuating allelic effects. In their symmetric larly when pleiotropy is considered, because it takes an model of G ϫ E interactions, all alleles have essentially extraordinary number of discrete alleles to approxi- the same mean and variance of effects. With this ex- mate even a two-dimensional continuum (Turelli 1985; treme symmetry assumption, even slight fluctuations can Wagner 1989). Models with discrete alleles also pre- maintain indefinitely many alleles at an arbitrary num- clude a particular multilocus genotype that produces ber of loci. However, the essential interchangeability the highest fitness under all conditions. By implicitly allowing such genotypes, Via and Lande (1987) con- of the alleles implies that there will be essentially no ϫ correlation between the phenotypes produced by a cluded that G E interactions could not maintain sta- given genotype across unrelated environments (i.e., two ble polygenic variation. Another critical assumption is that the temporal and environments chosen at random from the distribution spatial scales of fluctuating allelic effects are sufficiently of environments responsible for maintaining variation; small, relative to the timescale of selection, that we can Gillespie and Turelli 1989, 1990; Gimelfarb 1990). average over these fluctuations to approximate the al- Genetic variation that shows so little consistency of ef- lele-frequency dynamics with deterministic differential fects would severely limit the resemblance between par- equations. Both the linkage equilibrium and averaging ents and offspring across different environments. approximations were made by Gillespie and Turelli Below we explore the consequences of allowing ap- (1989), and we explore their validity numerically with preciable differences in the mean effects of different temporal fluctuations in allelic effects (and sex-depen- alleles. We show that under simple forms of spatial and dent allelic effects). We conjecture that more highly temporal variation in allelic effects, the conditions for autocorrelated temporal fluctuations would maintain the maintenance of variation become much more re- less variation (Gillespie and Guess 1978), whereas a strictive than those indicated by Gillespie and Turelli coarser spatial variation can maintain more variation (1989). Nevertheless, a surprisingly simple necessary (Barton and Turelli 1989; Barton 1999). condition for the maintenance of variation emerges. Our analyses show that balancing selection can main- Our weak-selection approximations apply to a broad tain variation at loci for which the intensity of balancing class of selection regimes in which balancing selection selection exceeds the strength of stabilizing selection. acts on the loci that contribute to trait variation. In With pleiotropy, this follows from sufficiently strong particular, we show that the approximate dynamics ob- balancing selection. In general, we find multiple alterna- ϫ tained for average allele frequencies under G E inter- tive stable equilibria, but these tend to produce similar actions and stabilizing selection are very similar to those mean phenotypes and levels of variation. With fluctuat- arising from pleiotropic balancing selection. ing allelic effects, stable polymorphism requires suffi- The consequences of sex-dependent allelic effects, as ciently large fluctuations in the effects and sufficient extensively documented by Mackay, Langley, and their independence of the fluctuations across loci. The restric- collaborators (e.g., Lai et al. 1995; Long et al. 1996; tiveness of the conditions is illustrated by the fact that Nuzhdin et al. 1997; Gurganus et al. 1998; Wayne and sex-dependent allelic effects cannot maintain stable poly- Mackay 1998; Vieira et al. 2000; Dilda and Mackay genic variation. Although fluctuations of allelic effects 2002), are approximated by a special case of the model that are extreme enough to maintain variation signifi- for spatial variation. Contrary to the expectation from cantly limit the consistency of genotypic effects, this lack single-locus analyses that such sex-dependent effects of consistency is apparent only if the genotypes are as- may promote the maintenance of variation, we show sayed across the entire range of environments responsi- Balancing Selection on Polygenes 1055 ble for maintaining variation. This may reconcile the poly- frequency dynamics can be described by ץ .morphism conditions with experimental observations ⌬ ϭ piqi ln w ץ pi 2 pi MODELS AND APPROXIMATE ANALYSES ϭ Spiqi ͑␣2 Ϫ Ϫ ␣ Ϫ␪͒ ϩ i (pi qi) 2 i(z ) o(S) (3) We analyze in turn pleiotropic balancing selection, 2 G ϫ E with spatial variation and complete mixing, ϫ (Wright 1937). (The first equation above is exact for sex-dependent allelic effects, and G E with temporal one locus; the approximation is the calculation of mean variation. The connection uniting these alternative sce- fitness for the one-locus genotypes.) Assuming weak narios is that in the weak-selection limit, they lead to selection, we can approximate (3) by essentially identical allele-frequency dynamics and hence similar stability properties for equilibria. This is dpi ϭ Spiqi ͑␣2 Ϫ Ϫ ␣ Ϫ␪͒ i (pi qi) 2 i(z ) . (4) somewhat surprising, since temporal G ϫ E leads to dt 2 stochastic fluctuations in allele frequencies whereas pleiotropic balancing selection, spatial variation, and Note that loci other than i enter these dynamics only sex-dependent allelic effects are wholly deterministic through their contribution to z, and this is true for all (but see Gillespie and Turelli 1989 for motivation of of the models we consider. Note also that the allele- this deterministic approximation and our results below frequency dynamics depend on the allelic effects only ␣ ϭ␤ Ϫ␥ Ϫ␪ϭ ͚n ␣ ϩ for numerical support). We start with the simplest deter- through i i i and z 2 iϭ1pi i ͚n ␥ Ϫ␪ 2 iϭ1 i . Because the scale of measurement of our ministic model to illustrate our stability analyses and ␪ then apply essentially the same analyses to “averaged” trait is arbitrary, can absorb any constants that enter the determination of the mean phenotype (such as the versions of more complex models involving environ- ␥ ment- or sex-dependent allelic effects. We support our i and the contributions of monomorphic loci not considered in our analyses). Thus, we are free to choose average-based analytical approximations with exact ␤ ␥ ␣ ϭ␤ Ϫ␥ multilocus numerical analyses and also use numerical any values for the i and i that satisfy i i i. analyses to explore the properties of simultaneously stable Without loss of generality, we assume that ␤ ϭ␣ ␥ ϭϪ␣ alternative equilibria. i i/2 and i i/2 for all i, (5) Pleiotropic balancing selection: Let Bi and bi denote so that the alleles at locus i.Weletpi,t denote the frequency of ϭ Ϫ n Bi in generation t and set qi,t 1 pi,t. We assume ϭ ␣ Ϫ z ͚ i(pi qi). (6) that selection is sufficiently weak and linkage sufficiently iϭ1 loose that we can ignore linkage disequilibrium. We assume diploidy and random mating. Let ␤ (␥ ) denote As shown initially by Wright (1935) from an approxi- i i mation like (4) (cf. Bulmer 1971), stabilizing selection the additive contribution of Bi (bi) to the trait of interest. ␣ ϭ␤ Ϫ␥ ␣ will generally eliminate additive polygenic variation (see We set i i i, so that i denotes the average effect of a substitution at locus i (Falconer and Mackay 1996, Bu¨rger and Gimelfarb 1999 for a recent review). To maintain variation, we assume that the loci experience Chap. 7). (Table 1 provides a glossary of notation.) As- suming no dominance or epistasis for the trait, the popu- balancing selection of some sort. The simplest such mechanism is overdominance, but our analysis also cov- lation mean and additive genetic variance in generation t are ers cases in which additive effects on fitness are linear functions of allele frequencies. This may be a good n n ϭ ␤ ϩ ␥ ϭ ␣2 approximation for a wide range of models of negative zt 2͚(pi,t i qi,t i) and VA,t 2͚ i pi,tqi,t . (1) iϭ1 tϭ1 frequency dependence, especially if allele frequencies are not perturbed too far from equilibrium. We assume constant Gaussian stabilizing selection on We assume that the relative contributions to fitness this trait with optimum ␪ and strength S, so that the fit- Ϫ Ϫ from pleiotropic effects are 1 siqî, 1, and 1 sipî for ness assigned to genotypes producing mean phenotype Ͻ Ӷ ϭ Ϫ BiBi, Bibi, and bibi, respectively, with 0 si 1, qî 1 G (averaged over nongenetic sources of variation) is pˆ , and 0 Ͻ pˆ Ͻ 1 for all i. These fitnesses lead to a w(G) ϭ exp(Ϫ(S/2)(G Ϫ␪)2); this produces both domi- i i stable equilibrium at pî, a fixed parameter in the model. nance and epistasis for fitness. For weak selection, we For definiteness, we assume that these pleiotropic fitness can approximate w(G) by a linear function of S.Inthe effects are multiplicative across loci and that this pleio- weak-selection limit, the population’s mean fitness is tropic selection acts before stabilizing selection in the life cycle, with both affecting viability. However, these as- ϭ Ϫ S ϩ Ϫ␪ 2 ϩ w 1 (VA (z ) ) o(S), (2) 2 sumptions are irrelevant in our weak-selection approximation. With weak selection, we can, like Bulmer (1973) where o(S) denotes a quantity that vanishes faster than and Gillespie (1984), superimpose the pleiotropic S does as S → 0. At linkage equilibrium, the allele- overdominant selection on the trait-induced selection 1056 M. Turelli and N. H. Barton

TABLE 1 Glossary of repeatedly used notation

Symbol Usage (relevant equation in the text)

Bi One of the two alleles at locus i (bi is the other), (1) K Fraction of the total genetic variance attributable to mean effects of genotypes, (44) S Strength of stabilizing selection, (2)

VA Additive genetic variance, (1) pi,t Frequency of Bi in generation t, (1) pˆi Equilibrium frequency of Bi under balancing selection alone, (7) qi Frequency of bi, (1) si Intensity of balancing selection at locus i, (7) ␣2 vi si/( i S) in the model of pleiotropic balancing selection, (7), or squared coefﬁcient of variation of the effect of substitution at locus i in the model of G ϫ E, (25) z Average trait value in the population, (1) ͚ ␣ Ϫ z*i Contribution to the mean phenotype from all loci but i, j϶i j(pj qj), (8) ␣ i (Mean) effect of a substitution at locus i, (1) and (25) ␤ ␥ i ( i) Additive contribution of Bi (bi) to the trait, (1) and (25) ⌬ Departure of the population mean from the optimum, z Ϫ␪, (7b) ⌬ f Under pleiotropic balancing selection, the amount by which the population would depart from the optimum in the absence of stabilizing selection, (23c) ␦ ⌬ ⌬ ␣ i normalized by the (mean) effect of a substitution at locus i, i.e., / i , (7) ␪ Optimum trait value under stabilizing selection, (2) ␳ ij The correlation between the substitution effects at loci i and j, (28a)

to approximate the allele-frequency dynamics by selection is eliminated and the population mean is at ⌬ϭ␦ ϭ the optimum ( i 0). Condition (9) creates a dpi ϭϪS ͑Ϫ␣2 Ϫ ϩ ␣ Ϫ␪ ϩ si Ϫ ͒ piqi i (pi qi) 2 i(z ) 2 (pi pî) system of linear equations for the p that will generally dt 2 S i ␣2 have a unique solution for a fixed set of polymorphic ϭ S i ͑ Ϫ Ϫ ␦ Ϫ Ϫ ͒ piqi (pi qi) 2 i 2vi(pi pî) , (7a) loci. We first focus on the stability of fully polymorphic 2 equilibria at which all n loci satisfy (9), but we show be- where low that precisely the same stability conditions emerge s ⌬ whenever two or more loci are polymorphic. Conditions v ϭ i , ␦ ϭ , ⌬ϭz Ϫ␪. (7b) i ␣2 i ␣ i S i for the feasibility of fully polymorphic equilibria are dis- ␣ cussed below, along with boundary equilibria at which Given that a deviation from the optimum of i reduces some or all of the loci are fixed. ␣2 fitness by S i /2, vi quantifies the intensity of balancing As shown in appendix a, stability of the fully polymor- selection at locus i relative to stabilizing selection. To phic equilibrium is determined solely by the vi. To main- make the stability analysis more transparent, we can tain a stable polymorphism, balancing selection must rewrite (7a) as be sufficiently strong relative to stabilizing selection dp 1 1 (Bulmer 1973). The stability of the fully polymorphic i ϭϪ ␣2 ͑ Ϫ ϩ Ϫ␪ ϩ Ϫ ˆ ͒ S i piqi pi ␣ (z*i ) vi(pi pi) , (8) dt 2 i equilibrium depends on the eigenvalues of the Jacobian matrix A ϭ (a ), with ϭ ͚ ␣ Ϫ ij where z*i j϶i j(pj qj) denotes the contribution to ץ the mean phenotype from all loci but i. ϭ dpi ϭϪ ␣2 ϩ (΂ ΃ Spiqi i (1 vi) (10a ץ aii Stability of fully polymorphic equilibria: From (7a) we see pi dt that each locus can fall into one of three possible equilib- ϭ ϭ and ria: pi 0, pi 1, or pi, satisfying ץ ˆ Ϫ Ϫ ␦ dpi ϭ 2vipi 1 2 i a ϭ ΂ ΃ ϭϪ2Sp q ␣ ␣ for i ϶ j, (10b) i i i j ץ pi Ϫ (9) ij 2(vi 1) pj dt ␦ (note that i depends on all of the allele frequencies), evaluated at allele frequencies that satisfy (9). The fully ␦ with vi and i defined in (7b). [Because of the simple polymorphic equilibrium is locally stable if all of the form (3) for our approximate dynamics, only point equi- eigenvalues of this matrix have negative real parts. In libria can occur; Bu¨rger 2000, Appendix A3.] As ex- general, it is difficult to calculate the eigenvalues. Never-

pected, the polymorphic equilibrium (9) becomes pˆi if theless, the stability conditions can be determined be- vi is very large and becomes 1/2 if pleiotropic balancing cause of symmetries imposed by our model. We consider Balancing Selection on Polygenes 1057

ﬁrst a completely symmetric model, as discussed by selection of intensity S␻ acts toward an optimum ␪␻, Bulmer (1973), for which all of the eigenvalues and independently across a set of characters, labeled ␻, i.e., the equilibrium allele frequencies can be explicitly de- w(G) ϭ Exp[Ϫ͚␻(S␻/2)(G␻ Ϫ␪␻)2] (corresponding to termined. multiplying the Gaussian selection across characters). ␣ ϭ Suppose that the loci are interchangeable with i Following the arguments leading to (7a), we obtain ␣ ϭ ˆ ϭ ˆ ϭ , si s, and pi p; then vi v for all i and the equations ˜ dpi ϭ ST ͑ Ϫ Ϫ ␦ Ϫ Ϫ ͒ (9) for the equilibrium allele frequencies have a unique piqi (pi qi) 2˜i 2v˜i(pi pˆi) , (17a) solution: dt 2 ␪ϩ Ϫ ␣ϩ ␣ where ϭ ϭ 2 (2n 1) 2v pˆ pi p . (11) 2␣(2n Ϫ 1 ϩ v) ͚␻ ␻␣2␻␦ ␻ ␻ Ϫ␪␻ ˜ ϭ ͚ ␣2 ␦˜ ϭ S i, i, , ␦ ϭ z , ϭ si . ST ␻S␻ i,␻, i i,␻ ␣ v˜i In this symmetric case, the stability matrix A has all ST i,␻ ST diagonal elements equal and all off-diagonal elements (17b) equal. A has only two distinct eigenvalues, The polymorphic equilibria can still be represented by ␦ ␦˜ ␭ ϭϪ ␣2 Ϫ ␭ ϭϪ ␣2 ϩ Ϫ (9) with vi replaced by v˜i and i replaced by i . 1 Spq (v 1) and 2 Spq (2n v 1), (12) However, the stability conditions are more complex than those for the one-dimensional model, because the ␭ Ϫ ␭ where 1 has multiplicity n 1. Obviously, 2 is always stability-determining matrix A in (10) is replaced by ␭ negative, but 1 is negative if and only if ϭϪ ϩ aii STpiqi(1 v˜i) (18a) v Ͼ 1. (13) and Thus, as Bulmer (1973) found by assuming that a ϭϪ2p p ͚␻S␻␣ ␻␣ ␻ for i ϶ j. (18b) z ϭ␪, s ϭ␣2S is the lower bound on the intensity of ij i j i, j, balancing selection relative to stabilizing selection that Because of the summation in (18b), the signs of the

must be exceeded to produce a stable polymorphism. eigenvalues of (18) do not depend solely on the v˜i . Essentially the same constraint on vi arises for the gen- Unlike the one-character model in which at most one Ͻ eral model (7a). locus is expected to be polymorphic with vi 1, for ϭ The necessary and sufficient conditions derived in multiple characters with v˜i 0 for all i and equal allelic appendix a for stability are that either effects, the number of stably polymorphic loci can be as large as the number of traits (Hastings and Hom v Ͼ 1 for all i, (14) i 1989) or larger with strong selection (Gimelfarb 1992). Ͻ or one locus (locus 1, say) has v1 1, but this locus We return to this result when we consider sex-depen- obeys dent allelic effects. Stability, feasibility, and positions of alternative equilibria: 1 v Ͼ 1 Ϫ . (15) Next, we consider equilibria for the one-character model 1 ϩ n Ϫ (1/2) ͚iϭ2(1/(vi 1)) in which some loci are monomorphic. Several complexi- For large numbers of polymorphic loci, the sum in ties arise due to the possible simultaneous stability of the denominator is large, and so this condition is barely multiple equilibria with different numbers of polymor- different from the simpler sufficient condition (14). phic loci and fixation of either Bi or bi at the monomor- Indeed, as shown in appendix a, a necessary condition phic loci. First, consider the conditions for polymorphic for stability is equilibria to be feasible. The conditions will depend on Ͼ whether vi 1 (recall that at most one stably polymor- ϩ ϩ Ͼ ϶ Ͼ (vi 1)(vj 1) 4 for all i j, (16) phic locus can violate this). If vi 1, (9) implies that 0 Ͻ p Ͻ 1 only if so that condition (14) is not far from being both neces- i sary and sufficient. Conditions (14) and (15) can be ϩ ␦ Ϫ ␦ Ͼ 1 2 i , 1 2 i . understood from Wright’s (1935) result that in the vi max ΂ ΃ (19a) 2pî 2qî absence of balancing selection, i.e., v ϭ 0 for all i,at i Ͻ most one locus is expected to be polymorphic in the If vi 1, feasibility requires Ͼ weak-selection limit. At loci with vi 1, balancing selec- ϩ ␦ Ϫ ␦ Ͻ 1 2 i , 1 2 i . tion is strong enough to maintain polymorphism. Our vi min ΂ ΃ (19b) 2pî 2qî weak-selection analysis indicates that at most one such Ͼ ϾϪ ⌬ϭ locus can be polymorphic with 1 vi 1. Unless 0, (19a) constrains at least all but one of ⌬ Multiple characters: The model readily generalizes to vi to exceed 1 by an amount that depends on . Con- multiple characters, but the resulting stability condi- versely, if stability is achieved with one locus satisfying Ͻ tions involve an important difference that illuminates vi 1, (19b) puts an upper bound on this vi that must our sex-dependent model. We suppose that stabilizing be satisfied along with the lower bound given by (15). 1058 M. Turelli and N. H. Barton ⌬ϭ ␣ Ϫ ␣ ϩ ␣ Ϫ Ϫ␪ Overall, these feasibility conditions for polymorphisms ͚ i ͚ i ͚ i(pi qi) . (22) ʦ⍀ ʦ⍀ ʦ⍀ and the conditions described next for stability of fixa- i 1 i 0 i p ⌬ tion equilibria require allele frequencies that make Substituting expression (9) for the equilibrium allele very small. frequencies and rearranging, we find that ϭ Consider an equilibrium at which pi 0 for all i in ⍀ ϭ ⍀ Ͻ Ͻ ⍀ ⌬ the set 0, pi 1 for i in 1, and 0 pi 1 for i in p . ⌬ϭ f , (23a) In this case, the stability matrix A can be partitioned 1 ϩ 2C into blocks corresponding to the fixed and polymorphic where loci, because ␣ v (pˆ Ϫ qˆ ) ϭ ϶ ⍀ ⍀ ⌬ ϭ ͚ ␣ Ϫ ͚ ␣ ϩ ͚ i i i i Ϫ␪ aij 0 for i j if i is in either 0 or 1 , (20a) f i i , (23b) ʦ⍀ ʦ⍀ ʦ⍀ Ϫ i 1 i 0 i p vi 1 whereas and ␣2 ϭ S i Ϫ Ϫ ␦ ʦ ⍀ 1 aii (2pîvi 1 2 i) for i 0 , (20b) ϭ . 2 C ͚ (23c) ʦ⍀ Ϫ i p vi 1 and The population mean would depart from the optimum ⌬ ␣2 by f in the absence of stabilizing selection returning ϭ S i Ϫ ϩ ␦ ʦ ⍀ aii (2qîvi 1 2 i) for i 1 . (20c) the trait toward the optimum. [Without stabilizing selec- 2 tion, the terms in the final summation in (23b) reduce ␣ Ϫ ⌬ Thus, the eigenvalues governing the stability of the fixed to i(pî qî).] In this sense, f represents a natural ʦ ⍀ ʜ ⍀ ␭ ϭ loci (i.e., i 0 1) are simply i aii , and the resting point of the system under balancing selection stability conditions for the subsystem of polymorphic alone. Stabilizing selection generally reduces this devia- ʦ ⍀ ϭ ϩ loci (i.e., i p) are just (14) and (15). Equations 20b tion by a factor B 1/(1 2C) (as noted in appendix Ͻ ⍀ and 20c show that the stability conditions for the fixed b, the factor is negative if vi 1 for one of the i in p, loci are but we ignore this special case). Thus, B is a cumulative measure of the strength of stabilizing selection relative 1 ϩ 2␦ v Ͻ i if p ϭ 0, (21a) to balancing selection. As noted above, we generally i ˆ i Ͼ 2pi expect vi 1 at stably polymorphic loci. For any fixed lower bound on the v , (23c) shows that as the number and i of polymorphic loci increases, the population mean will Ϫ ␦ Ͻ 1 2 i ϭ converge to the optimum by slightly perturbing the poly- vi if pi 1. (21b) morphic allele frequencies away from pˆ as described 2qî i by (9). Hence, the conditions for the stability of the fixed equi- The stability conditions for the full system, including libria are complementary to the feasibility conditions fixed and polymorphic loci, are detailed in appendix b. Ͼ (19a) for the polymorphic equilibria with vi 1. The The qualitative conclusion is that, for a wide range of Ͼ implications of (21) can be seen by assuming, without parameter values, loci with vi 1 can be stably polymor- ⌬ϭ Ϫ␪ Ͻ loss of generality, that z is negative and all of phic and loci with vi 1 are generally monomorphic. ␣ the i are positive. In this case, increasing pi at each Moreover, although alternative equilibria may be simul- locus moves the population mean closer to the opti- taneously stable, they generally produce mean pheno- ϭ mum. Then, inequalities (21) with vi 0 imply that, types very near the optimum. These generalizations are whenever possible, the multilocus system will equilibrate illustrated by our numerical examples below, which also ⌬ ʦ ␣ so that | | is less than mini ⍀0{ i/2}. Inequalities (21) suggest that the alternative equilibria produce similar ⌬ Ͼ imply that | | is even smaller with vi 0. equilibrium levels of genetic variation. Because alternative multilocus equilibria will gener- Consequences of G ϫ E with spatial variation and ally produce different values of ⌬, and hence different complete mixing: Next we consider a deterministic ␦ i for each locus, conditions (19) and (21) do not pre- model that involves only stabilizing selection on the clude a locus from having stable alternative fixation and trait, but allows for environment-specific allelic effects, polymorphic equilibria (cf. Hastings and Hom 1990). which can produce balancing selection at individual In particular, if vi is only slightly above one, the locus loci (Gillespie and Turelli 1989). Following Levene can be stably polymorphic at an equilibrium with ⌬ very (1953), we assume that each environment contributes near 0, but stably monomorphic at equilibria with larger a constant proportion to the random-mating pool that |⌬|. This is illustrated numerically below. forms the next generation of zygotes. In our weak-selec- Now consider ⌬ at equilibria. Assuming as above that tion limit, this means that we can simply average the ϭ ʦ ⍀ ϭ ʦ ⍀ Ͻ Ͻ pi 0 for i 0, pi 1 for i 1, and 0 pi 1 for equations that emerge in each environment, weighting ʦ ⍀ i p, we have each environment by its fractional contribution to the Balancing Selection on Polygenes 1059 ϭ ͚ Ϫ ␣ next generation (cf. Gillespie and Langley 1976). Let where E(z*i ) j϶i(pj qj) j denotes the average contri- ␤ ␥ i,k ( i,k) denote the effect of Bi (bi) in environment k. bution to the mean phenotype from the loci other than ␤ Ϫ␥ ϭ ϭ With weak selection, as in (4), we can approximate the i. It is easy to see that Cov( i i , z*i ) 0if˜cij 0 for allele frequency dynamics in this environment by all i ϶ j. Gillespie and Turelli (1989) assumed this and found that interlocus correlations did not affect dpi ϭϪSpiqi Ϫ Ϫ ␤ Ϫ␥ 2 ( (pi qi)( i,k i,k) their polymorphism condition. We show below that this dt 2 conclusion depends critically on their symmetry assump- ϩ ␤ Ϫ␥ Ϫ␪ 2( i,k i,k)(z )). (24) tion concerning interlocus correlations. Analysis of fully polymorphic equilibria: At equilibrium, We assume that S and ␪ remain fixed across environ- ϭ ϭ each locus must satisfy pi 0, pi 1, or ments. Averaging over environments, we define 1 ␣ [E(z*) Ϫ␪] ϩ␣2d˜ ϩ Cov(␤ Ϫ␥, z*) 2 ϭ Ϫ i i i ii i i i E(␤ Ϫ␥) ϭ␣ and Var(␤ Ϫ␥) ϭ v ␣ . (25) pi . i i i i i i i ␣2 ϩ 2 i (1 vi) ␣ (30) Thus, the effect of a substitution at locus i has mean i ␣2 and variance is vi i , so that vi is the square of the coeffi- Note that increasing d˜ii, corresponding to raising the cient of variation of the substitution effect. We demon- variance of effect for allele Bi relative to the variance strate that these vi play the same role in the stability for bi, decreases the equilibrium pi, consistent with the analysis of this model as do the vi defined by (7b) for general principle that selection in variable environ- the pleiotropy model. Averaging over environments, as ments tends to favor more homeostatic genotypes (Gil- done in Gillespie and Turelli (1989), we obtain lespie 1974). The stability of the fully polymorphic equilibrium depends on the eigenvalues of the Jacobian dpi ϭϪSpiqi ͑Ϫ␣2 ϩ Ϫ ϩ ␣ Ϫ␪ ϭ i (1 vi)(pi qi) 2 i[E(z) ] matrix A (aij) with elements dt 2 dp ץ ϩ 2 Cov(␤ Ϫ␥, z)͒. (26) ϭ i ϭϪ ␣2 ϩ (΂ ΃ Spiqi 1(1 vi) (31a ץ i i aii pi dt As discussed after Equation 4, we can absorb constants ␪ ␤ ϭ␣ that enter E(z) into , so we assume E( i) i/2 and and ␥ ϭϪ␣ ϭ ץ (E( i) i/2, without loss of generality. Thus, E(z ͚ Ϫ ␣ ϭ dpi ϭϪ ␣ ␣ ϩ ϶ i(pi qi) i . Rearranging (1), we have aij ΂ ΃ 2Spiqi i j(1 ˜cij) for i j, (31b) p dtץ n n j ϭ Ϫ ␤ Ϫ␥ ϩ ␤ ϩ␥ z ͚(pi qi)( i i) ͚( i i), (27) ϭ ϭ i 1 i 1 evaluated at allele frequencies that satisfy (30), with ˜cij ␤ Ϫ␥ ϭ ␤ ␥ as defined in (28a). If Cov( i i, z*i ) 0, the terms where the i and i have environment-specific values. ␤ Ϫ␥ ˜cij in (31b) vanish and the stability conditions are pre- Hence, the term Cov( i i , z) that enters (26) and the analyses below depends on the scaled covariances, cisely those for the pleiotropy model, (14) and (15). In this case, the G ϫ E model reduces to the pleiotropic ˜cij and d˜ij , defined by balancing selection model (apart from d˜ii , which does ␤ Ϫ␥ ␤ Ϫ␥ ϭ ␣ ␣ ϵ ␳ ␣ ␣ √ ϭ Cov( i i , j j) ˜cij i j ij i j vivj (28a) not affect stability of polymorphic equilibria) with pî 0.5 at all loci. Following the argument in appendix a, and it is easy to see that in general the stability properties ␤ Ϫ␥ ␤ ϩ␥ ϭ ␣ ␣ ␳ Cov( i i , j j) d˜ij i j , (28b) of (31) depend only on the vi and the ij defined in (28a). In general, positive correlations across loci are desta- ϭ ␳ where ˜cii vi and ij denotes the correlation of substitu- bilizing in the sense that larger v are needed to achieve ϭ ˜ ϵ i tion effects at loci i and j. Note that ˜cij dij 0 for all stability with ␳ Ͼ 0 than with ␳ ϭ 0. The destabilizing ϶ ij ij i j if either the allelic effects at different loci fluctuate effect is dramatic. The necessary condition for stability, independently or we impose the symmetry constraints, ␤ ␤ ϭ ␥ ␥ ϭ ␥ ␤ ϭ ␤ ␥ analogous to (16), is Cov( i, j) Cov( i, j) Cov( i, j) Cov( i, j) for ϶ all i j. Gillespie and Turelli (1989) assumed the (v ϩ 1)(v ϩ 1) Ͼ 4(1 ϩ␳√v v )2 for all i ϶ j. (32) ␤ i j ij i j latter. Under the less restrictive assumptions that Cov( i, ␤ ϭ ␥ ␥ ␥ ␤ ϭ ␤ ␥ j) Cov( i, j) and Cov( i, j) Cov( i, j), we have With ␳ Ͼ 0 and ␳2 Ͼ 3/4, (32) cannot be satisfied for ˜ ϵ ˜ ϭ ␤ Ϫ ij ij dij 0 for all i and j. In particular, dii Var( i) three or more loci. The general stability conditions can ␥ ˜ ϭ ␤ ϭ ␥ Var( i), so that dii 0 if Var( i) Var( i). Separating be explicitly obtained following the procedure given the terms in (26) that depend on locus i, we have in appendix a, but they seem too complicated to be informative. However, the qualitative effects of correla- dpi ϭϪSpiqi ͑␣2 ϩ Ϫ ϩ ␣ Ϫ␪ i (1 vi)(pi qi) 2 i[E(z*i ) ] tions across loci can be seen under the symmetry as- dt 2 ϵ ␳ ϵ ␳ sumptions vi v and ij . In this case, a feasible fully ϩ ␣2 ϩ ␤ Ϫ␥ ͒ 2 i d˜ii 2 Cov( i i , z*i ) , (29) polymorphic equilibrium is stable if and only if 1060 M. Turelli and N. H. Barton

1 v Ͻ 1 ϩ 2e ,ifp ϭ 0, and v Ͻ 1 Ϫ 2e ,ifp ϭ 1. ␳Ͻ1/2 and v Ͼ . (33) i i i i i i 1 Ϫ 2␳ (36) ␤ Ϫ␥ Hence, polygenic variation can be stably maintained As expected, positive values of Cov( i i , z*i ) and d˜ii ϭ under this model of G ϫ E interactions only if the loci promote the stability of pi 0. experience at most moderate positive correlations Properties of polymorphic equilibria: One of our central among their fluctuating allelic effects, and the variance motivations for allowing significant differences in the in effects is sufficiently large. As illustrated by our analy- mean effects of alleles was to determine whether appre- sis of sex-dependent allelic effects, with only two envi- ciable heritable variation could be maintained by G ϫ ␳ ϭ ronments, | ij | 1 and polymorphism condition (32) E that would provide persistent selection response. We cannot be satisfied. have shown that maintaining variation requires a suffi- Stability, feasibility, and position of alternative equilibria: ciently large coefficient of variation of the allelic effects ϭ ʦ ⍀ ϭ Consider an equilibrium with pi 0 for i 0, pi 1 and sufficient independence of the fluctuations across ʦ ⍀ Ͻ Ͻ ʦ ⍀ for i 1, and 0 pi 1 for i p. First note that if loci. At least two biologically interesting questions fol- ␤ Ϫ␥ ϭ ϭ Cov( i i , z*i ) 0 and d˜ii 0, we can use the results low. First, how similar are the phenotypes of various for the pleiotropic balancing selection model with the relatives, for instance, parents and offspring, and sec- ϭ additional constraint pî 0.5 for all i. This generally ond, how variable are the phenotypes produced by spe- simplifies the analysis. For instance, the stability condi- cific genotypes across the range of environments re- tions for the fixed loci reduce to sponsible for maintaining the variation (cf. Yamada Ͻ ϩ ␦ ʦ ⍀ Ͻ Ϫ ␦ ʦ ⍀ 1962; Gillespie and Turelli 1990; Gimelfarb 1990). vi 1 2 i ,ifi 0 , and vi 1 2 i ,ifi 1 , The second question is answered more easily than the (34) first, because the similarity of relatives will depend on ␦ ϭ Ϫ␪ ␣ where i [E(z) ]/ i . Numerical examples of pleio- the similarity of their environments. Even if this is tropic overdominance presented below, which assume known, the correlations between relatives will depend ϭ pî 0.5, illustrate the approximate equilibria and dy- on additional parameters, which do not enter the poly- namics of this model. morphism conditions, that describe the covariance of ␤ Ϫ␥ ϶ New phenomena appear with Cov( i i , z*1 ) 0 the fluctuating effects of alleles within and across loci. ϶ and d˜ii 0. As noted above, positive correlations across These parameters also enter the variance for the mean loci make stable polymorphisms more difficult to obtain phenotypes produced by specific genotypes across envi- and positive values of d˜ii tend to lower pi. Hence, we ronments. To illustrate this, we calculate the expected Ͼ expect that positive correlations between loci and d˜ii variance of the mean phenotype of a randomly drawn ϭ 0 will broaden the conditions for stability of pi 0. As genotype and then partition the equilibrium variance before, the stability matrix A can be partitioned into in mean phenotypes to quantify the consistency of geno- blocks corresponding to the fixed and polymorphic loci, typic differences across environments. because Let Gk(g) denote the average phenotype of a specific multilocus genotype g in a specific environment k (the ϭ ϶ ⍀ ⍀ aij 0 for i j if i is in either 0 or 1 , (35a) same analyses apply to both spatial and temporal varia- whereas tion). Under our additivity assumption, n S␣2 ϭ ϭ i Ϫ Ϫ ʦ ⍀ Gk(g) ͚Gi,k(g), (37) aii (vi 1 2ei) for i 0 , (35b) 2 iϭ1 and where Gi,k(g) denotes the contribution of the diploid genotype at locus i in environment k. Under the assump- ␣2 ϭ S i Ϫ ϩ ʦ ⍀ tions that lead to (24), the allele-frequency dynamics aii (vi 1 2ei) for i 1 , (35c) 2 depend on the moments of allelic effects only through the means and variances of the substitution effects. with ␤ Ϫ␥ ϭ␣ Thus, we had to specify only E( i i) i and ␤ Ϫ␥ ϭ ␣2 Cov(␤ Ϫ␥, z*) Var( i i) vi i . However, we will see that the vari- e ϭ␦ ϩ d˜ ϩ i i i . (35d) i i ii ␣2 ance of genotypic values depends separately on the vari- i ances and covariances of the allelic effects within and Thus, the eigenvalues governing the stability of the between loci. We assume that ʦ ⍀ ʜ ⍀ ␭ ϭ fixed loci (i.e., i 0 1) are simply i aii , and the ␤ ϭ ␥ ϭ ␣2 ␤ ␥ ϭ␳ ␣2 stability conditions for the subsystem of polymorphic Var( i) Var( i) ci i /2 and Cov( i , i) ici i /2, ʦ ⍀ (38) loci (i.e., i p) are determined by the eigenvalues of (31), which depend only on the parameters for the so that polymorphic loci. Equations 35b and 35c show that the ␤ Ϫ␥ ϭ ␣2 Ϫ␳ stability conditions for the fixed loci are Var( i i) ci i (1 i). (39) Balancing Selection on Polygenes 1061 ϭ Hence, in the calculations above, e.g., Equation 8, vi tal “patches” or years), it bears no simple relationship Ϫ␳ ␳ ci(1 i). [Note that i describes correlations between to heritability estimates, which also account for “micro- ␳ allelic effects within loci, whereas ij in (28a) describes cor- environmental” variation (i.e., phenotypic differences relations between substitution effects at different loci.] among genetically identical individuals within the same First, assume uncorrelated fluctuating allelic effects spatial or temporal macroenvironment). across loci, so that Our linkage equilibrium assumption and the defini- ␣ n tion of i imply that irrespective of correlations in fluc- ϭ Var[Gk(g)|g] ͚Var[Gi,k(g)|g], tuations within or among loci, iϭ1 n n Var {E[G (g)|g]} ϭ 2͚␣2p q . (45) E {Var[G (g)|g]} ϭ ͚E {Var[G (g)|g]}, (40) g k i i i g k g i,k iϭ1 iϭ1 Hence, for independent fluctuations across loci, where Eg denotes averaging over the distribution of ge- n ␣2 notypes in the population. A central feature of all ran- ͚iϭ1 i piqi K ϭ . (46) dom environment models is that Var[Gi,k(g)|g] depends n ␣2 ϩ n ␣2͑ Ϫ␳ Ϫ ͒ ͚ ϭ i piqi ͚ ϭ vi i (1/(1 i)) piqi on whether genotype g is homozygous or heterozygous i 1 i 1 at locus i (Gillespie and Turelli 1989). Using (38), The qualitative implications of (46) are most easily seen Var[G (g)|g] ϭ 2c ␣2 if g is homozygous for either allele i,k i i with exchangeable loci (i.e., ␣ ϭ␣, c ϭ c, ␳ ϭ␳, and at locus i, and Var[G (g)|g] ϭ c ␣2(1 ϩ␳)ifg is hetero- i i i i,k i i i p ϭ p), for which zygous at locus i. Thus, i pq(1 Ϫ␳) E {Var[G (g)|g]} ϭ 2c ␣2[1 Ϫ p q (1 Ϫ␳)] K ϭ , (47) g i,k i i i i i pq(1 Ϫ␳) ϩ v[1 Ϫ pq(1 Ϫ␳)] ϭ ␣2 1 Ϫ 2vi i ΂ piqi ΃. (41) and the stability criterion is simply v Ͼ 1. Equation 47 1 Ϫ␳ i implies that K decreases as ␳ and v increase and as ␳ Note that this depends on both vi and i. With indepen- p departs from 0.5. Hence, for stable equilibria, K is dent fluctuations across loci, we have maximized when ␳ϭϪ1, v ϭ 1, and p ϭ 0.5. At this point, K ϭ 0.5. However, when the within-locus effects n 1 ϭ ͚ ␣2 Ϫ are uncorrelated, as the between-locus effects are as- Eg{Var[Gk(g)|g]} 2 vi i ΂ Ϫ␳ piqi ΃. (42) iϭ1 1 i sumed to be, K Յ 0.25. To understand the implications of (42), we need a In general, positive between-locus covariances reduce scale-independent quantification of this expected within- K because the numerator remains constant but Eg{Var genotype variance. By analogy to broad-sense heritabil- [Gk(g)|g]} in the denominator increases. The effects of ity, we can define an index for the stability of environ- these covariances depend not only on the covariances ␤ Ϫ␥ ␤ Ϫ␥ ment-dependent genetic effects as the fraction of the of substitution effects, i.e., Cov( i i, j j)asde- total genetic variance (across both genotypes and envi- scribed by (28), but also on the covariances between ␤ ␤ ronments) attributable to the mean effects of different the individual alleles at each locus, i.e., Cov( i, j), ␥ ␥ ␤ ␥ genotypes. In general, averaging over both environ- Cov( i, j), and Cov( i, j). Nine different expressions ments and genotypes, we have for Cov{[Gi,k(g)|g], [Gj,k(g)|g]} are generated by the three genotypes at each locus. To illustrate the quantita- ϭ ϩ Var[Gk(g)] Varg{E[Gk(g)|g]} Eg{Var[Gk(g)|g]}, (43) tive effects, we focus on the completely symmetrical where, as indicated, the inner expectations on the right- case explored by Gillespie and Turelli (1989) and hand side are taken over environments and the outer Gimelfarb (1990) with expectations are taken over genotypes. The first term ␤ ␤ ϭ ␥ ␥ ϭ ␤ ␥ Cov( i , j) Cov( i , j) Cov( i , j) is the variance of the mean genotypic values (i.e., the ␳ √ ␣2 ␣2 “main effect” of genotypes) and the second is the aver- ϭ ␤ ␥ ϭ B ci i cj j Cov( j , i) . (48) age across-environment variance for individual geno- 2 types. We define the “consistency” of these genotypes as As discussed following Equations 28, this implies that Var{E[G (g)|g]} Cov(␤ Ϫ␥, ␤ Ϫ␥) ϭ 0 for all i ϶ j, so that the allele K ϭ k . (44) i i j j Var[G (g)] frequency dynamics are still approximated by (29) with k ϭ ␤ Ϫ␥ ϭ d˜ii Cov( i i , z*i ) 0. In particular, in the symmetri- K near 0 implies that the differences among mean ef- Ϫ␳ ϭ cal case with ci(1 i) v for all i, the polymorphic fects are small relative to the standard deviations of equilibrium is stable whenever v Ͼ 1. For exchangeable genotypes’ effects across environments, and K near 1 loci satisfying (48), (47) is replaced by implies relatively large mean effects (e.g., K ϭ 1 with Ϫ␳ constant allelic effects). Because K focuses on parti- ϭ pq(1 ) . K Ϫ␳ ϩ Ϫ Ϫ␳ ϩ Ϫ ␳ tioning the variance of genotypic means within and pq(1 ) v[1 pq(1 ) (n 1) B] among “macroenvironments” (e.g., spatial environmen- (49) 1062 M. Turelli and N. H. Barton ␳ Thus, for any positive B, K approaches 0 for large num- the qualitative conclusion that sex-dependent, additive bers of loci (see Gimelfarb 1990 for an analogous re- allelic effects cannot maintain stable polygenic varia- sult). The implications of these upper bounds on K for tion. From our sexes-averaged approximation, we ex- the maintenance of variation by G ϫ E interactions are pect that at most one locus can remain stably polymor- considered in the discussion. phic under weak selection. In fact, however, several Sex-dependent allelic effects: A special case of this numerical examples described below indicate that up multiple-environment model approximates allele-fre- to two loci may be stably polymorphic for loose linkage. quency dynamics with sex-dependent allelic effects. In This shows that our averaging approximation misses this case, the two sexes are the alternative “environ- some of the subtleties of sex-dependent selection, while ments.” As first argued by Haldane (1926) and demon- accurately capturing its inability to maintain variation strated rigorously for one locus by Nagylaki (1979), at many loci. The stable two-locus polymorphisms we the dynamics of weak, sex-dependent viability selection find are reminiscent of Hastings and Hom’s (1989, can be approximated by simply averaging the fitnesses 1990) results concerning pleiotropic effects on two char- of each genotype over the two sexes. This is equivalent acters. A more careful analysis of sex-dependent allelic to averaging the allele-frequency dynamics as in (26), effects requires distinguishing the allele frequencies in but now each random variable takes on only two values. the two sexes, so that at linkage equilibrium the stability This greatly simplifies and constrains the expressions analysis for n loci involves 2n variables rather than n. for the coefficients of variation and the correlations of This is discussed elsewhere. ␣ ␣ ϫ substitution effects across loci. For instance, if f,i ( m,i) Temporal variation and G E: Finally, we apply our denotes the effect of a substitution at locus i on females analyses to generalize the treatment by Gillespie and (males), we have Turelli (1989) of temporally fluctuating allelic effects. Our approximation is based on averaging over the distri- ␣ Ϫ␣ 2 ϭ f,i m,i . bution of allelic effects to approximate the stochastic vi ΂␣ ϩ␣ ΃ (50) f,i m,i dynamics by a set of deterministic equations identical to those obtained for G ϫ E with spatial variation. As The condition v Ͼ 1 requires that ␣ and ␣ have i f,i m,i noted below, this approximation relies on weak selec- different signs. Thus, if we use the convention that each tion. Ultimately, the usefulness of our approximations B denotes the allele that increases the trait value in i depends on their ability to predict the maintenance of females, the polymorphism condition v Ͼ 1 implies that i variation with biologically plausible levels of selection each B must decrease the trait in males. By considering i and environmental fluctuation. We present numerical the symmetrical model with ␪ϭ0, it is easy to see, results below, suggesting that our approximate polymor- however, that v Ͼ 1 cannot suffice to maintain polymor- i phism conditions are surprisingly accurate. Our deter- phism. The multilocus recursions for the allele frequen- ministic analysis does not address the fluctuations of cies depend separately on the fitnesses assigned to each allele frequencies inherent in stable polymorphisms genotype in males and females. When ␪ϭ0, our sym- maintained by temporal fluctuations. This is explored metrical selection model implies that altering the signs numerically below. of all of the allelic effects in one sex will not change As with spatial variation, under particular symmetry the fitnesses. Hence, for any assignment of allelic effects, assumptions concerning the interlocus correlations in identical dynamics must emerge if all of the signs of fluctuating allelic effects [see (28) and (29)], the deter- allelic effects in one sex are reversed. If the initial assign- Ͼ ministic approximation is precisely equivalent to the ment of effects satisfies vi 1 for all i, by reversing the pleiotropy model analyzed above. As with spatial varia- signs of effects in one sex, we get identical dynamics tion, the critical parameter governing the stability of but the new values of vi are the reciprocals of the old. polymorphism at each locus is just vi, the squared coef- The additional constraint required for stable poly- ficient of variation of the substitution effect at that locus. morphism involves the correlations in the fluctuat- Because we obtain identical approximations for tempo- ing effects across loci. Our convention of labeling the ral and spatial variation, we discuss the analytical ap- alleles so that Bi and Bj increase the trait value in females proximations only briefly. implies that The assumptions of this model are the same as with ␳ ϭ 1 for all i ϶ j (51) spatial variation, except that the allelic-effect parame- ij ␤ ␥ ters, i,t and i,t, vary across generations. Note that (25) (indeed, a two-valued bivariate random variable can allows for arbitrary correlation between the fluctuating have only correlations Ϯ1). Thus, constraint (32) im- effects of the two alleles at a locus. Within-locus correla- ␤ ␥ plies that a fully polymorphic equilibrium cannot be tions of i,t and i,t do not explicitly affect the dynamics, ␤ Ϫ␥ stable, although sex-dependent selection can readily because selection depends only on i,t i,t. However, maintain polymorphism at one locus (Kidwell et al. as discussed in the context of spatial variation, intralocus 1977). correlations can significantly affect the properties of the Below, we present numerical analyses that support genetic variation maintained, depending on the time- Balancing Selection on Polygenes 1063

scale of parameter variation. As before, the basic re- to be fixed within any one generation. With high levels cursion is of positive autocorrelation for allelic effects, K would remain high even when averages are taken over several ⌬ ϭϪSpi,tqi,t ͑Ϫ Ϫ ␤ Ϫ␥ 2 ϫ pi,t (pi,t qi,t)( i,t i,t) generations. Thus, temporal G E may maintain high 2 levels of genetic variation with consistent differences ϩ ␤ Ϫ␥ Ϫ␪͒ ϩ among genotypes over the timescale of reasonable ex- 2( i,t i,t)(zt ) o(S). (52) perimental analyses. This point is elaborated below. A central assumption of our analysis is that selection is weak; i.e.,SӶ 1. We also assume that the timescale of allele-frequency change is slower than the timescale NUMERICAL ANALYSES of the environmental fluctuations, for instance, that the Pleiotropic balancing selection: The number of pa- successive environments are independent or at most rameters in this model makes it impractical to explore “weakly autocorrelated” (Gillespie and Guess 1978). equilibria and dynamics systematically across the param- These assumptions allow us to (i) average the random eter space. However, the qualitative features of alterna- fluctuations over time and (ii) approximate the dynam- tive equilibria and dynamics are illustrated by the follow- ics of the discrete-generation stochastic model (52) by a ing two numerical examples. In both examples, we ϭ system of deterministic differential equations. [Because assume for simplicity that pî 1/2 at all loci. Loci with the leading term in (52) is proportional to S, the infini- extreme values of pî are less likely to be polymorphic tesimal variance in a diffusion approximation vanishes, because of the constraints associated with the feasibility leaving a deterministic limit (cf. Gillespie and Turelli conditions (19) and the increased influence of genetic 1989).] Taking the expectation of the right-hand side drift. We concentrate on observable quantities such as of (52) over the fluctuations in allelic effects, ignoring mean fitness, deviation of the trait mean from the opti- the higher-order terms and going to the continuous- mum, and genetic variance. time limit, we obtain the time-averaged, weak-selection Example 1—alternative equilibria: To illustrate the impli- approximation (26) used above to discuss spatial varia- cations of the stability and feasibility conditions, we first ␣ tion with complete mixing. We have not made any ex- consider an example in which the i and vi at 20 loci plicit assumptions about the temporal correlations of were drawn independently from gamma distributions the fluctuating parameters. However, our analysis im- (Table 2), and the optimum is ␪ϭ0. Because of this plicitly assumes that autocorrelations decay faster than symmetry, the equilibria come in pairs, the elements of the timescale of allele frequency change (1/S). which have ⌬ of opposite sign, produced by replacing Ϫ The approximate polymorphism conditions are pre- each pi by 1 pi. appendix c describes the procedure cisely those obtained for the spatial model considered for finding the alternative stable equilibria. Note that above. Again, the central parameters governing the sta- since all of the equilibria discussed have some mono- bility of polymorphisms are the vi, which describe the morphic loci, the polymorphic loci must produce a squared coefficient of variation of substitution effects mean near an optimum different from 0, which cannot ␳ at individual loci [see (25)], and the ij, which describe be achieved by multilocus heterozygotes or by any com- the correlations between substitution effects at differ- bination of homozygotes. This “effective optimum,” de- ␳ ϭ ␪ ent loci [see (28a)]. In particular, when ij 0, we noted eff,is expect that loci satisfying v Ͼ 1 will tend to remain i ␪ ϭ␪Ϫ ␣ ϩ ␣ polymorphic if the expected population mean is close eff ͚ i ͚ i ; (53) ʦ⍀ ʦ⍀ Ͻ i 1 i 0 enough to the optimum. Loci with vi 1 will generally not be stably polymorphic, and they will fix for alleles its values are given in Table 2. that produce an expected population mean very near ␪. For the parameter values in Table 2, there are seven As before, we predict from (32) that no stable multilocus pairs of distinct stable and feasible equilibria (or, more polymorphisms can be maintained for loci with large accurately, seven pairs of allele frequencies that satisfy ␳ positive ij. the constraints for stable and feasible equilibria given One important difference between the spatial and by our approximations); only the stable allele frequen- temporal models concerns the interpretation of the con- cies that produce negative ⌬ are shown. All of the equi- sistency index, K, defined by (44), and its relationship libria share similar properties: mean fitnesses are within to empirical observations. The key point, as made by 0.55S of each other, and the deviation from the opti- Gillespie and Turelli (1989, 1990), is that the tempo- mum is |⌬| Ͻ 0.373 (relative to a mean allelic effect of ral variation responsible for maintaining variation need ␣ϭ1 and a range of genotypic values |G | Ͻ 21.07). not be observable over a few generations. For instance, Barring the most extreme pair of equilibria (presented even though the “true” value of K, obtained by averaging in the first column of Table 2), mean fitnesses are within over all of the environments responsible for maintaining 0.0051S of each other, and the deviation from the opti- variation, may be quite small, the value observed in any mum is |⌬| Ͻ 0.021. The main differences among the one generation is one, since allelic effects are assumed equilibria are in the number of polymorphic loci and 1064 M. Turelli and N. H. Barton

TABLE 2 Multiple stable equilibria maintained by stabilizing selection on a single trait and overdominance

Allele frequencies (pi) at alternative stable equilibria (ei) ␣ Locus i vi e1 e2 e3 e4 e5 e6 e7 1 2.27458 0.048 0000100 2 1.92538 0.181 0111011 3 0.320365 0.220 1011011 4 0.054810 0.099 1100011 5 0.03597 0.486 1101000 6 0.004355 0.906 1111110 7 0.000191 0.792 1111111 8 0.000042 0.102 1111111 9 0.000053 3.364 1111111 10 0.761252 1.034 1 1 1 0.6995 0.7657 0.5248 0.5652 11 0.011769 3.528 1 1 0.5534 0.5174 0.5232 0.5022 0.5057 12 0.02935 2.720 1 0.9072 0.5315 0.5103 0.5137 0.5013 0.5034 13 0.21609 1.387 1 0.7457 0.5190 0.5062 0.5082 0.5008 0.5020 14 0.325396 1.038 1 0.7048 0.5158 0.5052 0.5069 0.5006 0.5017 15 0.0502681 3.936 1 0.6393 0.5108 0.5035 0.5047 0.5004 0.5011 16 0.037032 5.905 1 0.6132 0.5086 0.5029 0.5038 0.5004 0.5009 17 0.224072 1.906 1 0.6012 0.5078 0.5025 0.5034 0.5003 0.5008 18 4.50501 1.357 0.7321 0.5128 0.5010 0.5003 0.5004 0.5000 0.5001 19 3.70138 4.877 0.5260 0.5014 0.5001 0.5000 0.5000 0.5000 0.5000 20 7.28178 5.723 0.5108 0.5006 0.5000 0.5000 0.5000 0.5000 0.5000 ␪ eff 2.8141 0.4862 0.0384 0.0430 0.0573 0.0053 0.0141 ⌬Ϫ0.3728 Ϫ0.0206 Ϫ0.0016 Ϫ0.0005 Ϫ0.0007 Ϫ0.0001 Ϫ0.0002

VA 41.2927 43.5909 43.6135 43.6161 43.6157 43.6166 43.6165 L/S 122.3500 121.802 121.797 121.797 121.797 121.797 121.797 Ϫ Ϫ Ϫ Ϫ Ϫ Ϫ L⌬/S 0.0695 2.1 ϫ 10 4 1.2 ϫ 10 6 1.3 ϫ 10 7 2.4 ϫ 10 7 2.1 ϫ 10 9 1.4 ϫ 10 8

LV/S 20.6464 21.7954 21.8068 21.8080 21.8079 21.8083 21.8083 LB/S 101.3340 100.0070 99.9904 99.9891 99.9893 99.9889 99.9889 ␣ Allelic effects i and relative overdominance vi for 20 loci were drawn from independent gamma distributions with means 1, 2, respectively, and variance 4. The range of the trait is |z| Ͻ 21.07, and the maximum possible ϭ ␪ϭ variance is VA 48.11. With an optimum at 0, there are 14 stable equilibria (according to our approximate stability criteria); only 7 are shown here, since the other 7 differ by only a sign change (and replacing the pi Ϫ by 1 pi). The last four rows give total genetic load and its three components [as deﬁned in (56)], scaled relative to S.

␣ Ϫ in the combinations of fixed loci. As expected from ⌬ Ͻ i(vi 1) ʦ ⍀ | | for all i p . (54) the stability conditions for fixation equilibria (21), the 2 polymorphic loci that differ among equilibria tend to ⌬ be those of small effect, and the different combinations Hence, for these loci to remain polymorphic, | | must ʦ⍀ ␣ Ϫ of polymorphic and monomorphic loci all bring the be less than mini p{ i(vi 1)/2}. Conversely, if the loci ⍀ population mean very close to the optimum. This ex- in 0 are to remain stably fixed, (21a) requires plains the similarity in overall fitness properties of the ␣ Ϫ ⌬ Ͻ i(1 vi) ʦ ⍀ different equilibria. Given that selection is simply tend- | | for all i 0 . (55) 2 ing to climb toward local fitness optima, it is not clear whether equilibria with higher mean fitness will tend With multiple loci contributing to the character, (54) to be reached preferentially. and (55) ensure a very close approach to the optimum. We can gain some insight by considering the feasibil- Because the deviation from the optimum is small, allele ity conditions for the polymorphic equilibria, (19), and frequencies at polymorphic loci with significant effects ⌬ the stability conditions for the fixation equilibria, (21). are close to pî . For small , the allele frequencies should ⌬Ͻ ␣ Ͼ 1 Ϫ ϩ Ϫ For definiteness, we assume that 0 and i 0 for be near ⁄2(1/(1 vi)) pî(vi/(vi 1)), which is a com- all i, so that directional selection will tend to increase promise between the unstable equilibrium under stabi- ⍀ the pi. As before, we let p denote the set of polymorphic lizing selection (first term) and the stable equilibrium ⍀ ϭ ϭ loci and 0 the set of loci with pi 0. With pî 1/2, under balancing selection (second term). For pî near 0.5, ͚␣2 1 ف the feasibility condition (19) requires that for the loci the genetic variance will be ⁄2 i , where the sum is ⍀ ϭ Ͼ in p to avoid fixation at pi 1, over those loci for which vi 1, and which are hence Balancing Selection on Polygenes 1065 polymorphic at ⌬ϭ0. For example, the model of Ta- maintained; on average, 6.8 loci were polymorphic [with ble 2 gives a genetic variance that varies across equilibria standard deviation (SD) 2.3]. In 12% of cases, one of Ͻ from 41.2927 to 43.6166, whereas the value when all these polymorphic loci had vi 1 as allowed by our Ͼ ϭ loci with vi 1 are polymorphic with pi 0.5 is 43.6166. stability condition. Conversely, 40% of the monomor- ⌬ Ͻ If deviates slightly from zero, then loci with small phic loci had vi 1; these loci had small allelic effects effects fix; however, such loci have little effect on the relative to the deviation from the optimum, so that ␣ Ϫ Ͻ ⌬ genetic variance. i(vi 1) 2| |. Typically, many alternative equilibria ␣ In general, the mean relative fitness of the population were stable for any given set of { i, vi}: on average, 17.7, is less than one because of stabilizing selection and with a range from 2 to 178. However, these equilibria balancing selection at the individual loci, which contrib- had very similar properties. This is because the equilib- utes an amount denoted LB to the genetic load. The ria differ in whether loci with small effects [namely, ␣ Ϫ Ͻ ⌬ stabilizing-selection component of the load can be parti- i(vi 1) 2| |] are fixed for 0 or 1. Every allowable tioned into the portion attributable to departures of set of these loci leads to almost the same (small) devia- the population mean from the optimum, denoted L⌬, tion from the optimum, and this set contributes very and variance in the population around the population little to the overall genetic variance. mean, denoted LV. In our weak-selection limit, the con- Over our replicates, the mean deviates from the opti- tributions to the load are additive, i.e., mum by an average ⌬ϭ0.17 (SD 0.28). This can be ␣ϭ ϭ Ϫ Ϸ ϩ ϩ compared with a mean allelic effect 1 and with an L 1 W LV L⌬ LB , (56a) average genetic variance of 31.1 (SD 39.9). The loss of with mean fitness caused by this deviation is negligible (mean 0.06S, SD 0.17S) compared with the genetic load due ϭ SVA ϭ ␣2 to variation around the optimum (mean 15.5S,SD LV S ͚ i piqi , (56b) ʦ⍀ 20.1S) and due to balancing selection (mean 59.1S,SD 2 i p 73.6S). In the great majority of cases, most of the genetic S⌬2 L⌬ ϭ , (56c) load is due to the perturbation of loci away from their 2 equilibrium under balancing selection. and Example 2—response to changes in selection: Next we consider the consequences of varying the intensity of stabi- n n ϭ 2 ϩ 2 ϭ Ϫ 2 lizing selection and the position of the optimum. We LB ͚si(pîqi qîpi ) ͚si(pî pi) iϭ1 iϭ1 explore how these alter the amount of variation main- n tained and the portions of the genetic load attributable ϩ ϭ ϩ ͚sipîqî LBS LBB , (56d) to departures of the mean phenotype from the optimal iϭ1 phenotype [as described by L⌬ (56c)], genetic variation where LBS in (56d) denotes the segregation load from about the mean [LV (56b)], and loss of fitness under balancing selection attributable to allele frequencies be- balancing selection caused by stabilizing selection per- ing perturbed by stabilizing selection from their bal- turbing allele frequencies away from their balancing ancing-selection equilibria, and LBB denotes the equilib- selection equilibria [LBS in (56d)]. In general, we expect rium segregational load under balancing selection that a moderate number of loci under balancing selec- Ͼ alone. From Table 2, we see that L⌬, the loss in mean tion might affect a specific trait (10–100 with si 0, say). fitness due to ⌬, is much smaller than that due to stabiliz- There will be a distribution of strengths of balancing ing selection (LV), which in turn is smaller than LB, the selection, si , and for each si , a distribution of allelic ef- ␣ segregational load. fects i that may be correlated with the si . In Figure 1, ␣ This numerical procedure was repeated with100 ran- we show a bivariate distribution of ( i, si) for 100 loci, dom choices of the set of allelic effects and strengths where the si were chosen from a gamma distribution ␣ of balancing selection { i, vi} at 20 loci. Although the with mean 0.01 and coefficient of variation (CV) 1, and ␣ quantitative outcomes depend on the arbitrary distribu- for each si , i was chosen from a gamma distribution ϩ ϭ tions chosen for these parameters, the qualitative fea- with mean 0.05 5si and CV 1 and then made nega- tures seem robust. In all but one case, the results were tive with probability 0.5 (i.e., a gamma distribution re- similar to those shown in Table 2. [In the one excep- flected around 0). There may also be many loci that tional case, the largest allelic effect was much greater affect the trait, but are not under balancing selection. ␣ ϭ than the rest: 1 17.8, compared with a maximum of These are not considered because they are expected to 1.78 among the remainder (results not shown). Thus, be fixed and thus will not affect the properties we dis- a single polymorphism was maintained by overdomi- cuss. For any given optimum, there may be many stable nance at this locus, with the remaining loci fixed for “1” equilibria, as illustrated in Table 2. We circumvent this alleles.] In the remaining discussion, we discard this by considering equilibria that produce a specific depar- outlier. ture from the optimum, ⌬. Although alternative equilib- In the other 99 cases, multiple polymorphisms were ria may exist even for fixed ⌬ (i.e., monomorphic loci 1066 M. Turelli and N. H. Barton

Figure 2.—Additive genetic variance, VA, as a function of the intensity of stabilizing selection, S. We assume that ⌬ϭ 0 as in Figure 1A. VA declines inversely with the strength of stabilizing selection (A). However, the load due to variation ϭ around the optimum, LV (S/2)VA, is almost constant for a ␣ Figure 1.—Values of i and si for 100 loci separated by wide range of S (B). lines indicating their stable equilibria. A assumes ⌬ϭ0 and B assumes ⌬ϭ0.05; both assume S ϭ 1. With ⌬ϭ0, there ␣ are only two stability regimes; loci with { i, si} above the parab- in the optimum. There will be an immediate increase ola must be polymorphic, while those below the parabola can in the “natural” deviation, to ⌬ Ͼ 0. Polymorphic allele ⌬ϭ f fix for 0 or 1. With 0.05, there are seven stability regimes, frequencies will adjust rapidly, and some loci will fix. shown in B, as described in the text. Overall, the new deviation from the optimum will be ⌬ ϩ *f /(1 2C*) [see (23)], which may be very much ⌬ may fix at either 0 or 1), the statistics we discuss have smaller than f. Note that the natural resting point will ⌬ ⌬ unique values because they depend only on polymor- shift from f to *f because some initially polymorphic phic loci. loci have fixed, which also decreases C* from its initial Suppose that the set of fixed loci is such that the value, C. In this new selection regime, the loci depicted natural resting point of the system coincides with the in Figure 1A are scattered over seven stability regimes ⌬ ϭ optimum; i.e., f 0 [see (23b)]. In this case, the condi- as shown in Figure 1B. Reading from left to right, we Ͼ tions for feasible and stable polymorphism are just vi see: (i) loci that could fix at 0 or 1 under the old and Ͼ ␣2 ␣ ␣ Ͻ 1orsi S i ; that is, polymorphic loci must have ( i, new positions of the optimum; (ii) loci with i 0 that si) that lie above the parabola in Figure 1A. Because the were originally fixed for 0 or 1 are now forced to fix ϭ ϭ mean coincides with the optimum and pî 1/2 at all for pi 0, increasing the trait value; (iii) loci that were ϭ of the loci, the equilibrium allele frequency at each lo- polymorphic are now forced to fix for pi 0; (iv) loci 1 2 ϭ ͚ ʦ ␣ cus is 1/2, and the genetic variance is VA ⁄2 i ⍀p i . that were polymorphic, and remain so; (v) loci that were ␣ Ͼ ϭ Figure 2A shows the genetic variance, VA, as a function polymorphic with i 0 are forced to fix for pi 1; of the strength of stabilizing selection: it decreases in- (vi) loci that were originally fixed for 0 or 1 are now ⌬ϭ ϭ versely with S. These calculations assume that 0as forced to fix for pi 1; and finally, (vii) loci that were in Figure 1A. Figure 2B shows LV, the load due to varia- fixed for 0 or 1 and remain so. tion around the optimum, as a function of S. Initially, Figure 3 shows the short-term deviation from the opti- ⌬ ⌬ LV increases linearly. However, as polymorphic loci start mum, , as a function of f, the difference between the to fix, LV decreases and remains almost constant for a natural resting point and the optimum. Figure 3A shows wide range of S. It is not clear how general this pattern the results on the original scale, and Figure 3B presents ␣ is, since it depends on the joint distribution of ( i, si). the same results as a log-log plot. In these calculations, Now, consider the short-term response to a decrease we assume that the polymorphic loci are those found Balancing Selection on Polygenes 1067

Figure 3.—The deviation of the mean ⌬ ⌬ from the optimum, , as a function of f. ⌬ϭ⌬ The line f is also plotted. (A) The original scale and (B) a log-log scale are shown. This plot depends on the assumption that the polymorphic loci are those identiﬁed with ⌬ϭ0 in Figure 1A.

assuming ⌬ϭ0, as in Figure 1A. The question addressed most variation is lost. The load due to variation around is how those specific polymorphic loci are expected to the optimum, LV, decreases as variation is lost, but this respond to changes in the optimum. As the optimum is compensated almost exactly by the load due to per- changes, all of the polymorphic allele frequencies adjust turbing the allele frequencies away from their equilibria ⌬ to produce a new value , as described by Equation 23a. under balancing selection, LBS. The overall load, L, ⌬ ⌬ As f increases, there is remarkably little change in , barely changes as the mean is perturbed slightly. ⌬ provided that f does not approach the maximum that Long-term responses of the mean and the additive can be compensated by shifts in polymorphic allele fre- variance to changes in the intensity of selection or the ⌬ ϭ quencies (corresponding to f 2.38 in this example, position of the optimum are complex and depend on 1.85⌬ ف ⌬ see Figure 3A). For small deviations, is only 0.01 f a large number of parameters describing the underlying (as determined by a least-squares fit of the log-log results genetics and the history of population size changes. in Figure 3B). The main effect of a small perturbation, However, the qualitative behavior is roughly as follows. ⌬ f, is to reduce the genetic variance (Figure 4), as poly- If the strength of stabilizing selection increases, genetic morphic allele frequencies shift from 0.5 and eventually variation will be rapidly lost, as some polymorphic loci fix. The efficiency with which selection optimizes the become unstable (as illustrated in Figure 4); the time- ␣2 mean of a polygenic trait, despite underlying constraints, scale for this loss is set by S i . If stabilizing selection is seen in other models of this sort (Barton 1986, 1999). weakens again, then polymorphism will be recovered Using the same procedure described for Figure 3, much more slowly, since the particular alleles under Figure 5 shows how three components of load, L⌬, LV, balancing selection must arise by mutation. Our assump- and LBS, and the total load, L (all scaled by the intensity tion here is that the alleles would be lost for many plaus- of stabilizing selection S), increase as the mean is per- ible population sizes rather than retained at mutation- turbed from the optimum. We ignore the contribution selection equilibrium. This follows because the muta- LBB to the total load, described in (56d), because this tion rates relevant to these particular alleles under bal- does not vary with changes in the polymorphic allele ancing selection would be on the order of the per- frequencies. As expected from the close adjustment of the trait mean to the new optimum (Figure 3), the load due to deviations of the mean, L⌬, is negligible until

Figure 5.—Genetic load, L, and its components, scaled by ⌬ S, as functions of f. The total load is the solid line at the top, and the load due to deviations of the mean from the optimum Figure 4.—The genetic variance, VA, as a function of the is the thick solid curve that stays near 0. The portions of ⌬ deviation of f (defined by Equation 23b). This is based on the load attributable to variation around the optimum (short ϭ ⌬ ϭ the same assumptions as Figure 3. VA 0.072 when f 0 dashes) and perturbations of the allele frequencies from their and declines to zero when the deviation is so large that all overdominant equilibria (long dashes) change relative magni- ⌬ loci have fixed. tude as f increases. 1068 M. Turelli and N. H. Barton nucleotide mutation rate, say 10Ϫ8 or smaller. The chance allowable allele frequencies (explained below). Itera- that one such mutation at locus i will be fixed is twice tions were stopped when the sum of absolute changes Ϫ ␣2 the selection coefficient, si S i ; hence, the timescale of gamete frequencies fell below a specified threshold. for recovery in a population of effective size N is set by For all results reported here, with equilibrium allele ␮ Ϫ ␣2 ␮ 2N i(si S i ). Because i is likely to be of the order frequencies restricted to (0.01, 0.99), a threshold of of the per-nucleotide mutation rate, recovery of poly- 10Ϫ9 or 10Ϫ10 proved adequate (the smaller value was morphism could be very slow, unless the population is used with weaker selection). For all cases in which two large enough to retain the previously polymorphic al- or more loci remained polymorphic, an output file was leles through mutation-selection balance. For example, generated giving the parameter values and initial ga- ϭ with an average si 0.01 as assumed in the example of mete frequencies. This allowed us to test the adequacy Figure 1, N ϭ 106 and ␮ϭ10Ϫ8, the rate of recovery of of our stopping criterion by lowering the threshold polymorphism at each locus individually is 2 ϫ 10Ϫ4. value and determining whether the same approximate To make this argument in a little more detail, suppose equilibria were obtained. It also allowed us to reuse that a population with loci as illustrated in Figure 1 is specific allelic effects and initial frequencies with differ- initially under stabilizing selection with S ϭ 1. Balancing ent selection intensities, as discussed below. ϭ selection then maintains genetic variance VA 0.072, Each set of simulations involved specifying the follow- due to the pleiotropic effects of 36 polymorphic loci. ing: the number of loci; the number of replicate sets If this variation is lost as a result of strong stabilizing of allelic effects; the number of initial conditions for selection, which then relaxes to its original intensity, each set of allelic effects; the mean and CV of the allelic- ␤ ␤ ␥ ␥ the rate of increase in the number of polymorphic loci effect parameters, i,f, i,m, i,f, and i,m (for simplicity ف ͚ ␮ Ϫ ␣2 ϭ is 2N i i(si S i ) 0.008; thus, it will take 4500 gen- we assumed the same mean and CV for each); the inten- erations to return to the original 36 polymorphic loci. sity of stabilizing selection, S; the optimal trait value, ␪; Similarly, if the optimum changes, different responses the recombination rates between adjacent loci (assum- occur on different timescales. After the first phase of ing no interference); a threshold for minimum accept- response to a new optimum, in which allele frequencies able polymorphic allele frequencies (to avoid artifacts adjust to approach the new optimum (Figure 3) and associated with slow convergence to fixation, we set this some polymorphic loci fix, there will be a loss of genetic threshold at 0.01); and a threshold for the sum of the variation (Figure 4). We now expect a second phase, in absolute changes in gamete frequencies. For the sex- which the very many loci that may affect the trait but are dependent simulations, four independent pseudo-ran- not under balancing selection accumulate mutations, so dom, gamma-distributed deviates were chosen for each ϭ ␤ ϭ as to bring the mean back toward the optimum. After locus, denoted gi for i 1,...,4;andweset i,f g1, ⌬ ϭ⌬ϭ ␥ ϭϪ ␤ ϭ ␥ ϭϪ this phase, we expect f 0 to a very close approxi- i,f g2, i,m g3, and i,m g4. For sex-independent ␤ ϭ␤ ϭ ␥ ϭ␥ ϭϪ mation, and so the stability regimes return essentially to effects, we set i,f i,m g1 and i,f i,m g2.To those depicted in Figure 1A. A third, and much slower, reduce the dimensionality of the parameter space and phase now occurs, in which variation at loci under bal- facilitate investigating many sets of allelic effects, we ϭ ␣2 ancing selection and lying above the parabola, si S i , used five unlinked loci, chose 10 sets of random initial is recovered. allele frequencies for each set of allelic effects, assumed ␪ϭ Sex-dependent allelic effects: To test the accuracy of that each gi has mean 1, and set 0 for all calculations. our weak-selection approximations, calculations were Our assignment of sex-specific allelic effects implies that performed using the full multilocus, diploid gamete- the effects of substitutions have the same sign in both frequency recursions with selection, random mating, sexes. However, as explained above, with ␪ϭ0, identical ␤ ϭ ␥ ϭϪ and recombination. Our goal was to test the prediction results are obtained by specifying i,f g1, i,f g2, ␤ ϭϪ ␥ ϭ that sex-dependent allelic effects cannot maintain stable i,m g3, and i,m g4, so that the effects of substitu- multilocus polymorphisms, at least with weak selection. tions in the two sexes have different signs. Given that stable two-locus polymorphisms can be main- Several thousand sets of parameters were explored, tained under strong selection for sex-independent ef- all with unlinked loci. These led to four simple general- fects (Gimelfarb 1996; Bu¨rger and Gimelfarb 1999), izations: (i) there were no stable equilibria involving we performed comparable sets of simulations with sex- three or more polymorphic loci; (ii) sex-dependent al- dependent and sex-independent allelic effects. We started lelic effects facilitate stable two-locus polymorphisms; a set of simulations by choosing independent random (iii) for both sex-dependent and sex-independent ef- values for the allelic effects assigned to males and fe- fects, choosing the effects from a distribution with larger males. The effects were chosen using gamma-distrib- CV facilitates stable two-locus polymorphisms; and (iv) uted, pseudo-random numbers, as described below. For for sex-dependent effects, unlike sex-independent alle- each set of allelic effects and selection parameters, we lic effects, stable two-locus polymorphisms can be found generally used 10 randomly chosen initial conditions. even with extremely weak selection. We briefly describe We started at linkage equilibrium with each allele fre- some results supporting these generalizations. quency chosen from a uniform distribution over the Over thousands of sets of allelic effects, each run with Balancing Selection on Polygenes 1069

TABLE 3 tion can be seen by concentrating on initial conditions Effects of sex dependence, CV of allelic effects, and intensity and sets of allelic effects that produce stable two-locus of stabilizing selection on the occurrence of stable polymorphisms with S ϭ 0.2. Using 30 such sets of allelic two-locus polymorphisms with five-locus selection effects and initial conditions, we set S ϭ 0.02 and iterated to a new equilibrium. The results are shown in the Sex dependent Sex independent third row of Table 3. For sex-independent selection, of N a Polyb (total)c N Poly (total) the 30 sets that led to stable two-locus polymorphism with S ϭ 0.2, only 1 produced a stable two-locus polymor- ϭ ϭ CV 0.5, S 0.2 1000 7 (12) 2000 1 (3) phism with S ϭ 0.02. (As expected, that example had CV ϭ 1.0, S ϭ 0.2 1000 117 (315) 1000 12 (42) ␤ ϭ ϭ d one polymorphic locus with very large effects, 1 S 0.02 30 17 30 1 ␥ ϭϪ ϭ e e 5.19973 and 1 8.78685, and one with much smaller S 0.002 17 17 1 0 ␤ ϭ ␥ ϭϪ S ϭ 0.0002 17 17 effects, 2 0.319015 and 2 0.144994.) When these same allelic effects and initial conditions were used with a Number of sets of allelic effects generated. S ϭ 0.002, only the locus of large effect remained poly- b Number of those sets in which at least one stable two-locus polymorphic equilibrium was found when iterating from 10 morphic. In contrast, for sex-dependent effects, 17 of random initial frequencies. the 30 sets of allelic effects and initial conditions also c Total number of calculations (out of 10N) that produced produced a stable two-locus polymorphism with S ϭ stable two-locus polymorphisms. 0.02 (even though only a single initial frequency was d For S Յ 0.02, each of these sets of allelic effects produced ϭ used). Moreover, for all 17, a very similar two-locus poly- a stable two-locus polymorphism with S 0.2. For each one, ϭ ϭ we iterated from only one of the initial frequencies that led morphism was also reached with S 0.002 and S to a stable two-locus polymorphism with S ϭ 0.2. 0.0002. As noted above, these stable two-locus polymor- e Only sets of allelic effects and initial frequencies that pro- phisms obtained with sex-dependent allelic effects are ϭ ϭ duced stable two-locus polymorphisms with S 0.2 and S analogous to those found by Hastings and Hom (1989, 0.02 were used. 1990) when alleles pleiotropically affect two characters under stabilizing selection. Although sex-dependent allelic effects do produce stable two-locus polymorphisms, 10 sets of initial allele frequencies, no stable equilibria our results suggest that they cannot maintain stable poly- were found with more than two polymorphic loci. This genic variation, even with strong selection. is consistent with the numerical results of Bu¨rger and ϫ Gimelfarb (1999) for sex-independent effects (see Temporal variation and G E: As with sex-dependent their Table 1). Although stable three-locus polymor- allelic effects, we tested our polymorphism conditions, phisms can be obtained even with sex-independent ef- based on weak-selection, deterministic approximations, fects when recombination rates are low relative to selec- by performing exact multilocus iterations with tempo- tion (Gimelfarb 1996), such polymorphisms seem rally varying allelic effects. The joint distribution of the unlikely with loose linkage and weak selection. fluctuating allelic effects depends on many parameters. Table 3 presents numerical results that illustrate the As noted in our analytical approximations, intralocus effects of sex dependence, interlocus variation in allelic correlations between allelic effects do not affect the effects, and the intensity of selection. The fact that larger polymorphism conditions, but interlocus correlations CV for allelic effects produces more two-locus polymor- between substitution effects dramatically affect the lev- phisms is expected from the analytical work of Nagy- els of variation required to maintain polymorphism [see laki (1989) and Bu¨rger and Gimelfarb (1999), show- (33)]. To test our predictions concerning variances and ing that strong selection and significant asymmetries of interlocus correlations of substitution effects, we used effects across loci are required to maintain stable two- symmetry assumptions to simplify the model description locus polymorphisms with stabilizing selection on an and our predictions. For all of our simulations, we as- ␪ϭ ␤ ϭ␣ ␥ ϭϪ␣ additive trait. The effect of asymmetries and the effect sumed that 0, E( i) /2, E( i) /2, Var ␤ ϭ ␥ ϭ ␣2 ␤ ␥ ϭ ␥ ␤ ϭ of sex dependence are illustrated with S ϭ 0.2 in Table ( i) Var( i) v , Cov( i, i) 0, Cov( i, j) ␤ ␥ ϭ ␤ ␤ ϭ ␥ ␥ 3 by comparing results from CV ϭ 0.5 vs. CV ϭ 1 for Cov( i, j) 0, Cov( i, j) Cov( i, j), and no auto- sex-dependent vs. sex-independent allelic effects. In correlation in the effects across generations. The allelic each case, larger CV produces significantly more sets effects were chosen as multivariate pseudo-random, of allelic effects leading to two-locus polymorphisms Gaussian deviates with the appropriate mean and covari- (under Fisher’s exact test, P Ͻ 10Ϫ9 for sex dependence ance structure, which depends on only three parame- and P Ͻ 10Ϫ4 for sex independence). Similarly, for each ters: ␣, the mean effect of a substitution at each locus; CV, sex dependence facilitates two-locus polymorphisms v, the squared CV of substitution effects [see (25)]; and (P Ͻ 0.01 for CV ϭ 0.05, P Ͻ 10Ϫ9 for CV ϭ 1). ␳, the interlocus correlation in substitution effects [see The qualitative difference between sex-dependent and (28a)]. Our calculations assumed six unlinked loci. With sex-independent allelic effects with respect to maintain- these symmetry assumptions, our approximate polymor- ing stable two-locus polymorphisms under weak selec- phism criterion reduces to (33), namely ␳Ͻ1/2 and 1070 M. Turelli and N. H. Barton

Figure 6.—Stable equilibria under fluctuating allelic effects. (A and B) The mean allele frequency (across generations) and the standard deviation of allele frequencies for the “most polymorphic” and “least polymorphic” (explained in the text) of six diallelic loci. The average frequency of the least common allele is given for each locus. The dashed vertical lines indicate that the predicted critical values of the parameter varied. In A, stable polymorphism is expected only for values of v above the critical value; monomorphism is expected for smaller values of v. In B, stable polymorphism is expected only for values of ␳ below the critical value. v Ͼ 1/(1 Ϫ 2␳). This prediction is independent of ␣␳ϭ0.17. According to (33), all loci should remain sta- and S. bly polymorphic if v Ͼ 1.515 and all loci should become Even with these symmetry assumptions, no attempt monomorphic if v Ͻ 1.515. The critical value for v is will be made to present simulations spanning the entire indicated by the dashed line in Figure 6A. As predicted, parameter space. Instead, we provide illustrative exam- for v Յ 1.4, all loci become monomorphic, with mean ϭ ples, using biologically plausible parameter values for SD ϭ 0. Conversely, for v Ն 1.6, all six loci remain selection intensity and average allelic effects, which fo- polymorphic. For v ϭ 1.5, very near the predicted cus on our predicted critical values for v and ␳. Figure threshold value, we see that at least one locus has be- 6 shows the effects of varying either v or ␳, holding all come monomorphic, but at least one remains polymor- other parameters fixed. These simulations assume ␣ϭ phic. As expected given the high level of stochastic fluc- 0.7 and S ϭ 0.05 (corresponding to the canonical values tuations, the polymorphic loci always show considerable used in Turelli 1984 and many other articles to explore fluctuations in allele frequencies. To put the observed polygenic mutation-selection balance). To summarize SDs in perspective, note that if the allele frequencies the asymptotic behavior of the stochastically fluctuating fluctuated between 1 and 0 in an extremely rapid man- allele frequencies, we first iterated the recursions for ner so that the allele frequency is essentially 1 with prob- 500,000 generations starting with random initial allele ability p and 0 with probability 1 Ϫ p, we would observe frequencies and global linkage equilibrium. We then an average allele frequency of p and SD near the maxi- ran the recursions for an additional 500,000 genera- mum value, √p(1 Ϫ p). When v ϭ 1.5, the SD of the of this maximum %86ف tions, during which we calculated the mean and stan- most polymorphic locus is dard deviation of allele frequencies at each of the six value. In contrast, for v ϭ 1.6, the SD for the most loci. We report in Figure 6 the means and SDs of the polymorphic locus is 78% of the maximum, and this two loci whose average allele frequencies depart least ratio declines somewhat as v increases, despite the in- and most from 0.5. The former is called the “most poly- creasing intensity of the stochastic fluctuations. morphic locus,” and the latter, the “least polymorphic.” Figure 6B considers varying ␳ with v ϭ 1.5. Prediction To standardize the results, we report the average fre- (33) implies that all loci should remain stably polymor- quency of the less common allele. phic if ␳Ͻ0.167 (see the dashed line in Figure 6B), Figure 6A shows the consequences of varying v with and all loci should become monomorphic if ␳Ͼ0.167. Balancing Selection on Polygenes 1071

As predicted, all loci become monomorphic when ␳Ն trait, despite stabilizing selection. At each locus, poly- 0.18, and all loci remain polymorphic when ␳Յ0.16. morphism can be maintained provided that two condi- With ␳ϭ0.17, very near the predicted threshold, we tions are met. First, balancing selection must be stronger ϭ ␣2 Ͼ see that at least one locus has become monomorphic, than stabilizing selection [vi si/( i S) 1]; at most but at least one remains polymorphic. one polymorphic locus can violate this condition. Sec- As expected from the way the allele frequency dynam- ond, the net balancing selection must be stronger than ics in (29) depend on ␣2 and S, varying each of these the directional selection that arises when the trait mean parameters has a similar effect on polymorphisms. For deviates from its optimum. At equilibrium, the trait instance, if we set v ϭ 1.6, ␳ϭ0.17, and ␣ϭ0.07, the mean closely matches the optimum. The genetic vari- mean allele frequency at the most polymorphic locus ance is maintained by a set of loci that are highly poly- remains very close to 0.5 for S ϭ 0.01, 0.05, and 0.25, morphic and is approximately equal to half the sum of but the SD increases from 0.32 with S ϭ 0.01 to 0.45 squared allelic effects at these loci. At each polymorphic with S ϭ 0.25. This reflects the fact that, with stronger locus, the allele frequency is a compromise between the selection, allele frequencies respond faster to changing equilibrium favored by balancing selection and a slight selection forces produced by varying allelic effects. Simi- shift that brings the overall trait mean close to the opti- larly, if we set v ϭ 1.6, ␳ϭ0.17, S ϭ 0.05, and vary ␣2 mum. Our numerical results address the properties of from 0.09 to 2.56 (roughly a factor of 25, as with S alternative equilibria and the consequences of bouts of above), again the mean stays very near 0.5 while the SD directional selection for variation, but we do not reiter- increases from 0.30 to 0.45. ate our findings here. Overall, our simulations suggest that our approxima- How likely is it that genetic variation is maintained tions provide useful guidelines concerning the mainte- as a pleiotropic effect of balancing selection? We have nance of polygenic variation through fluctuating allelic little idea how many balanced polymorphisms there effects. As shown in Figure 6, the simulations switch might be, but it is plausible that variation is maintained from stable multilocus polymorphisms to complete fix- by selection at a substantial fraction of genes (at some ation near the predicted threshold values for v and ␳. thousands of loci in multicellular eukaryotes, say). There As the parameters near the threshold values, interlocus is then no difficulty in accounting for high levels of differences become greater and allele-frequency fluc- genetic variance in any particular trait. However, this tuations become more extreme. explanation faces two difficulties in explaining variation in most quantitative traits. First, balancing selection must be strong enough in total to counterbalance the DISCUSSION stabilizing selection acting on all traits. Roughly speak- Two basic classes of models explain the maintenance ing, we expect that the total strength of balancing selec- ͚ of stable polygenic variation: those that rely on mutation tion, isi, should be greater than the net load due to to maintain variation that would otherwise be largely variation of quantitative traits around their optima. Un- eliminated by selection and those in which selection fortunately, we do not know the magnitude of either of itself maintains variation. For a broad range of biologi- these quantities. If balancing selection is due to over- ͚ cally reasonable parameter values, mutation-selection dominance, then isi is proportional to the segregation balance models imply that the variation maintained will load, which could in principle be measured as a compo- generally be attributable to rare alleles at many loci nent of inbreeding depression. However, if frequency- (Turelli 1984). This remains true even for the most dependent selection predominates, it is hard to relate ͚ recent models of mutation-selection balance that con- isi to observable quantities. The net genetic load due sider both direct and pleiotropic selection (e.g., Zhang to deviation of traits under stabilizing selection from and Hill 2002). In contrast to this theoretical expec- their optima is still harder to estimate; indeed, it is hard tation, molecular studies suggest that variants at inter- even to define the number of traits under stabilizing mediate frequency contribute significantly to polygenic selection (though see Orr 2000). Despite these uncer- variation in natural populations (e.g., Mackay and tainties, however, it is at least possible that there is suffi- Langley 1990; Long et al. 2000). This provides one cient balancing selection to counterbalance stabilizing of the primary empirical motivations for our study of selection on very many traits. For example, selection ͚ ϭ alternative models for balancing selection, because such coefficients of 5% on 2000 loci would give isi 100. models generally lead to intermediate allele frequencies This could counterbalance genetic loads of a few per- at the polymorphic loci. We discuss in turn the results cent due to stabilizing selection on some thousands of from each of our models and then make some general independent traits. comments, comparing alternative mechanisms for the More naively, we can ask how much balancing selec- maintenance of variation. tion is required to maintain variation at a particular Pleiotropic balancing selection: Our analysis pro- locus that affects a single trait under stabilizing selec- duces straightforward conditions under which balanc- tion. If we assume a heritability near 0.5 and scale ge- ing selection can maintain variation in a quantitative netic and environmental variance to 1, we can ask how 1072 M. Turelli and N. H. Barton much balancing selection is needed to maintain a poly- mutation-selection balance. When allelic effects are equal, of the total genetic vari- the variance can increase dramatically when the mean %10ف morphism contributing ␣2 ance. From (1), we have i near 0.2. Hence, if stabilizing deviates above the optimum. This is because all the loci selection is on the order of S ϭ 0.05, we require balanc- near p ϭ 0 climb in frequency together, until a critical ϭ ing selection on the order of si 0.01. If stabilizing value is approached when some set of loci switch to the selection is often much weaker than assumed (King- alternative equilibrium with p near 1. Near this critical solver et al. 2001), the required intensity of balancing value, the equilibrium genetic variance can greatly ex- selection decreases. ceed that expected with the mean at the optimum (Bar- A second constraint is that episodes of directional ton 1986). However, with varying allelic effects and equi- selection on quantitative traits must not eliminate varia- librium allele frequencies, a single locus approaches tion at polymorphic loci. In our analysis, we assumed the critical value and then fixes, greatly reducing the stabilizing selection toward a constant optimum. In real- magnitude of the deviation from the optimum without ity, optima may vary, and so allele frequencies at the giving much increase in genetic variance. underlying loci will fluctuate. If directional selection is Fluctuating allelic effects: Under our models of tem- sufficiently strong for long enough, alleles will fix, and poral or spatial fluctuations, a necessary condition for variation will be regenerated only when lost alleles are the maintenance of polymorphism at a locus is that vi, recovered. We consider such a scenario in detail above, the squared coefficient of variation of the effects of a at the end of our numerical analyses of pleiotropic bal- substitution (across the distribution of environments), ancing selection. Clearly, if balancing selection is suffi- exceeds one. To address the biological plausibility of ciently strong, and if traits depend on very many loci, this condition, we first describe its mathematical impli- then the mean can be adjusted by small changes at each cations by specifying distributions for the substitution locus, avoiding fixations. Moreover, if selection varies effects. The range of implications can be illustrated by from place to place in a spatially subdivided popula- considering two particular distributions: Gaussian and tion, alleles can be retrieved by migration rather than Ͼ gamma. The stability condition vi 1 implies that the by mutation. Finally, it may be that balanced polymor- standard deviation of substitution effects exceed the phisms are usually transient [as seems to be the case mean. Under a Gaussian distribution, this implies that for inversions in Drosophila (Andolfatto et al. 2001) the sign of substitution effects at this locus must fre- and for human adaptations to malaria (Hamblin et al. quently change with environmental conditions. If we 2002)] rather than maintained for very long times (as, assume that the mean effect of a substitution is positive, for example, with incompatibility loci in flowering ⌽ Ϫ the probability of a negative effect will be ( 1/√vi), plants, or the human histocompatibility system in verte- ⌽ brates; Hughes 1999). This idea is related to the tran- where denotes the cumulative distribution function of the standard normal distribution. This probability sient maintenance of variation through fluctuating se- Ͼ ϭ lection on traits themselves, as discussed below. must be at least 0.16 for vi 1, it is 0.24 for vi 2, and Relation with mutation-selection balance: Combining it approaches 0.5 as vi increases. Such sign reversals Ͼ mutation with stabilizing selection leads to substantial are not necessary, however, because vi 1 can also mathematical complications. The polymorphic equilib- be achieved with a gamma distribution, which remains Ͼ ria are now given by the solution to a system of cubic positive. In this case, vi 1 puts a lower bound on the equations, and there may be multiple stable polymor- magnitude of fluctuations of the substitution effects. phic equilibria for a fixed set of polymorphic loci. In One way to quantify this is as the ratio of the 75th contrast, the model analyzed here gives a unique poly- percentile of the distribution of substitution effects to morphic equilibrium for any fixed set of polymorphic loci. the 25th percentile (which depends only on vi). This Ͼ ϭ However, the two cases are similar; indeed, they must be ratio must be at least 4.82 for vi 1, it is 13.0 for vi because mutation represents a small perturbation of 2, and it approaches infinity rapidly as vi increases (e.g., ϭ the model analyzed here and thus will give qualitatively it is 100 for vi 4.01). similar results (e.g., Karlin and McGregor 1972). Both Are such dramatic fluctuations in substitution effects stabilizing selection and mutation-selection balance plausible? The most relevant data concerning the fluc- generally have the property that the mean can be ad- tuating effects of individual loci are the quantitative trait justed to small changes in the optimum by slight changes loci (QTL)-based G ϫ E studies by Mackay and her col- in allele frequencies at many loci. In both models, many laborators (e.g., Gurganus et al. 1998; Vieira et al. 2000; combinations of fixed or nearly fixed loci can give stable Dilda and Mackay 2002). These studies estimate the equilibria, all of which produce a population mean very effects of individual QTL, which presumably correspond close to the optimum. to one or a small number of closely linked loci, over a Our numerical results suggest that with unequal al- range of environmental conditions, such as alternative lelic effects the properties of different equilibria be- rearing temperatures, heat shock, and starvation; they come more similar to each other than do those in the also document sex dependence. It is important to recog- case where loci are equivalent. The same may hold for nize, however, that the genotypes used in these analyses Balancing Selection on Polygenes 1073 are generally recombinant inbred lines derived from tions. Similar data are available for life span (Vieira et al. selection experiments or long-held laboratory stocks. 2000) and bristle number (Gurganus et al. 1998), but Hence, the variation described may not be representa- in these studies, the genetic variation originates from tive of variation in natural populations. Nevertheless, recombinant inbred lines derived from long-held labo- these studies demonstrate that QTL effects are generally ratory stocks. Again, these studies indicate the ubiquity sex or environment dependent. Scanning the data ta- of environment- (and sex-)dependent genetic effects. bles in these articles (see, for instance, Table 4 of Vieira However, they cannot tell us whether the variation segre- et al. 2000), examples can be found where statistically gating in natural populations satisfies the constraints significant marker effects within a sex vary by more than expected for stable G ϫ E -maintained polymorphisms. a factor of 10 or change sign depending on the rearing Many other recent studies demonstrate G ϫ E at the environment. Hence, these data seem broadly compati- level of either QTL (e.g., Shook and Johnson 1999) or ble with the levels of variation required to maintain whole genotypes (Shaw et al. 1995), but none provide polymorphism in our analysis. However, Mackay and col- data that allow us to estimate the relevant parameters. laborators clearly recognize another important caveat One possible argument against the role of G ϫ E in (e.g., Dilda and Mackay 2002, p. 1671). We do not know maintaining variation is that significant levels of additive whether the environments chosen for these laboratory variance are routinely found in laboratory populations, experiments are representative of the environmental including long-established stocks, experiencing relatively variation in nature that may be responsible for main- homogeneous and constant environments (Weigens- taining genetic variation. It seems reasonable, neverthe- berg and Roff 1996). For populations recently estab- less, to assume that the range of conditions in nature lished in the laboratory, two opposing forces affect VA: would fluctuate in many more ways than considered sampling, which tends to diminish variation, and re- in these experiments, with temperature, crowding, and laxed selection, which tends to maintain variation that food quality, for instance, all varying simultaneously in might be eroded under the more stringent selection time and space. expected in nature. To the extent that laboratory condi- Analyzing fluctuating allelic effects for individual loci tions minimize selection, levels of additive variance may is extremely difficult. A more traditional quantitative- approach a mutation-drift equilibrium that might be genetic approach is to consider the “consistency index,” considerably higher than that maintained by either mu- K (see Equation 44), the ratio of the variance of mean tation-selection or balancing selection alone (Turelli effects of genotypes to the total genetic variance, which et al. 1988). Because of the radical differences expected includes mean effects plus interaction terms related to in the selection regimes, data on variation from labora- G ϫ E. As noted above, the polymorphism conditions tory populations cannot preclude a central role for G ϫ constrain K to be quite small, especially when the fluc- E in maintaining variation in nature. tuations in effects across loci are positively correlated Sex-dependent allelic effects: We have shown for dial- (see Equations 47 and 49). Indeed, 0.25 is an upper lelic loci that sex-dependent additive allelic effects are bound for K if all fluctuating allelic effects, both within no more effective at maintaining stable polygenic varia- and across loci, are uncorrelated, but this bound quickly tion than is the classic sex-independent additive model falls to values on the order of 0.1 or less under more investigated by Wright (1935). (Although sex-depen- plausible assumptions. Relevant data appear in several dent effects can maintain variation at two loci whereas experimental studies of G ϫ E that partition the total sex-independent effects maintain variation at only one, genetic variance observed across genotypes and environ- this distinction is negligible in the context of under- ments into main effects of genotypes and interaction standing polygenic variation.) In contrast, sexually an- effects. For instance, Wayne and Mackay (1998) used tagonistic fitness effects can easily maintain single-locus three temperatures to study ovariole number and body polymorphisms (e.g., Kidwell et al. 1977). This distinc- size in mutation-accumulation lines of Drosophila melano- tion between the propensity of sex-dependent effects to gaster. Treating temperature and block as the environ- facilitate one-locus polymorphism but inability to main- mental variables (see their Tables 1 and 2), we see that tain polygenic variation is analogous to findings con- of the newly arising variation for cerning antagonistic pleiotropic effects on life histories %57ف for ovarioles genotypes (lines) plus interactions between genotypes (compare Rose 1982 and Curtsinger et al. 1994). and environments is attributable to mean effects of ge- Sex-dependent allelic effects have been extensively notypes. The comparable amount for body size is 42%. documented for several traits in D. melanogaster (sum- Given that these estimates come from a sample of newly marized in Dilda and Mackay 2002). Our analytical arising variation, not all of which would be expected to and numerical results indicate that such effects per se remain stably polymorphic because of G ϫ E, there is cannot account for the maintenance of polygenic varia- no reason to expect them to satisfy the constraints on tion for traits under stabilizing selection. We do not yet K described above. Nevertheless, they demonstrate that know how such sex-dependent effects will interact with mutation provides environment-sensitive variants that sex-dependent fitness regimes. could plausibly satisfy the G ϫ E polymorphism condi- Comparisons to alternative mechanisms: We have con- 1074 M. Turelli and N. H. Barton sidered models in which selection alone maintains quan- most of the variance in their model is contributed by titative variation. Many other models of this type have a single locus as it sweeps between near fixation for been proposed. For example, two-locus polymorphisms alternative alleles. This does not seem to be an adequate can be maintained by stabilizing selection on an additive explanation of polygenic variation. In the context of trait if allelic effects are sufficiently different, even when our pleiotropy model, adding balancing selection to selection is weak (Nagylaki 1989). When selection is these models might have rather little effect, because strong enough that linkage disequilibrium is significant, fluctuating selection would tend to eliminate the partic- polymorphism may be further facilitated (Gavrilets ular alleles that are required for balanced polymor- and Hastings 1994). However, Bu¨rger and Gimelfarb phism. Bu¨rger and Gimelfarb (2002) consider fluctu- (1999) used numerical investigations to show that when ating optima under stabilizing selection for moderate more than a few loci influence a trait, polymorphism numbers of diallelic loci. They demonstrate that consid- at multiple loci becomes much less likely; moreover, erably more variation can be maintained than expected the loci of smallest effect tend to be polymorphic, so under mutation-selection balance alone, but their re- that very little genetic variance is maintained. This pat- sults seem to depend on fairly extreme fluctuations in tern arises because with multiple loci the optimum can the position of the optimum relative to the width of the be closely matched by a homozygous genotype; selection stabilizing selection function (cf. Turelli 1988). If, as then acts against deviations from this optimal genotype. Kingsolver et al. (2001) have recently argued, stabiliz- With multiple traits, it is harder to match the trait mean ing selection is typically much weaker than usually as- to the optimum, and so relatively more polymorphism is sumed, the fluctuating-optimum hypothesis will be less expected: in Hastings and Hom’s (1989) model, there credible. can be as many polymorphic loci as there are traits Epistatic interactions have been widely documented under stabilizing selection. A serious criticism of these (e.g., Shook and Johnson 1999; Dilda and Mackay models is that they apply to only outcrossing diploids 2002) and some numerical work has suggested that epis- and so cannot account for quantitative variation in hap- tasis may help maintain polygenic variation (e.g., Gimel- loid or selfing organisms that do not contain heterozy- farb 1989). However, recent analytical work by Her- gotes. Surprisingly little is known about polymorphism misson et al. (2003) suggests that epistasis is likely to conditions in multilocus haploid models with recombi- lower rather than raise the additive variance maintained nation (e.g., Kirzhner and Lyubich 1997). On the ba- by mutation-selection balance. sis of Rutschman’s (1994) two-locus analysis, it is rea- Future directions: We have shown that pleiotropic sonable to conjecture that epistatic selection cannot balancing selection and temporally or spatially varying maintain variation in the absence of dominance inter- allelic effects can maintain stable polygenic variation, actions. In contrast, provided that balancing selection but sex-dependent allelic effects cannot. Recent studies acts through negative frequency dependence rather have championed varying selection pressures as a way to than through overdominance, the pleiotropic balancing explain the maintenance of alleles at intermediate fre- selection model we analyze is quite generally applicable. quencies (e.g., Waxman and Peck 1999; Bu¨rger and Fitness might depend on genotype frequencies through Gimelfarb 2002) and pleiotropic effects of deleterious a number of selective mechanisms mediated by quan- alleles on weakly selected traits (Zhang and Hill 2002) titative traits (see the theoretical analyses of Bulmer as a way to explain abundant additive variance attribut- 1974, Slatkin 1979, and Bu¨rger 2002 and the data able to rare alleles at many loci. These models and ours analyzed by Bolnick et al. 2003). However, it seems present a daunting challenge to experimentalists to esti- simplest to treat variation in some arbitrary trait as being mate the relevant parameters. As with molecular varia- due to the pleiotropic effects of balanced polymor- tion at individual loci, the explanation of polygenic vari- phisms, without detailing the causes of that balancing ation may well depend on the simultaneous action of selection. many alternative mechanisms, both across characters In the models just discussed, selection on the trait and across loci. Mutation-selection balance surely ex- remains constant through time and we focus on stable plains some of the variation we observe, but it is gener- equilibria. In contrast, Bu¨rger (1999), Waxman and ally expected to explain only the persistence of rare Peck (1999), and Bu¨rger and Gimelfarb (2002) have alleles (but see Slatkin and Frank 1990). Intermediate recently shown that a changing optimum can generate allele frequencies might be explained by either some substantially more genetic variance than would be gen- form of balancing selection, as we have discussed, or erated in a balance between mutation and static stabiliz- transient polymorphisms associated with fluctuating se- ing selection. Bu¨rger (1999) and Waxman and Peck lection (or hitchhiking effects from linked sites under (1999) assumed a continuum of allelic effects at each directional selection). locus, and it is unclear how far variation can be inflated Studies of molecular variation at and near individual with discrete alleles under their selection regimes. Kon- loci that contribute to polygenic variation may help drashov and Yampolsky (1996) demonstrate an in- unravel the relevant evolutionary forces. In particular, creased genetic variance under fluctuating selection in elevated levels of molecular variation in regions that a model with discrete alleles. However, at any one time, contribute to quantitative variation are expected if bal- Balancing Selection on Polygenes 1075 ancing selection has maintained alleles for very long a balance between mutation and stabilising selection. Genet. Res. 47: 209–216. times (Hudson et al. 1987). Such persistent polymor- Barton, N. H., 1990 Pleiotropic models of quantitative variation. phism and excess variability have been seen at self- Genetics 124: 773–782. incompatibility loci in plants and around the MHC in Barton, N. H., 1999 Clines in polygenic traits. Genet. Res. 74: 223–236. Barton, N. H., and M. Turelli, 1989 Evolutionary quantitative mammals (Hughes 1999). However, it is likely that se- genetics: How little do we know? Annu. Rev. Genet. 23: 337–370. lection on balanced polymorphisms fluctuates over Bolnick, D. I., R. Sva¨nback, J. A. Fordyce, L. H. Yang, J. M. Davis time, and such fluctuations will reduce neutral diversity et al., 2003 The ecology of individuals: incidence and implications of individual specialization. Am. Nat. 161: 1–28. over wide regions of the genome (with recombination Bulmer, M. G., 1971 The stability of equilibria under selection. rates comparable to the selection coefficient), whereas Heredity 27: 157–162. balancing selection is expected to increase diversity in Bulmer, M. G., 1973 The maintenance of the genetic variability of polygenic characters under heterozygote advantage. Genet. Res. only narrow regions (with recombination rates compa- 22: 9–12. rable to the mutation rate, ␮) and then only when the Bulmer, M. G., 1974 Density-dependent selection and character same allele is maintained for times of order 1/␮. Thus, displacement. Am. Nat. 108: 45–58. chromosomal inversions in Drosophila (Andolfatto Bu¨rger, R., 1999 Evolution of genetic variability and the advantage of sex and recombination in changing environments. Genetics et al. 2001) and polymorphisms that confer malaria resis- 153: 1055–1069. tance in humans (e.g., Sabeti et al. 2002) are associated Bu¨rger, R., 2000 The Mathematical Theory of Selection, Recombination, with reduced—not increased—variability, because they and Mutation. Wiley, Chichester, UK. Bu¨rger, R., 2002 On a genetic model of intraspecific competition are of relatively recent origin. and stabilizing selection. Am. Nat. 160: 661–682. These arguments are based on the expected effects Bu¨rger, R., and A. Gimelfarb, 1999 Genetic variation maintained of balancing selection at single loci. Balancing selection in multilocus models of additive quantitative traits under stabilizing selection. Genetics 152: 809–820. at many loci can in principle maintain enormous diver- Bu¨rger, R., and A. Gimelfarb, 2002 Fluctuating environments and sity at linked sites, because it maintains many genotypes the role of mutation in maintaining quantitative genetic variation. (defined by all possible combinations of selected al- Genet. Res. 80: 31–46. Caballero, A., and P. D. Keightley, 1994 A pleiotropic non- leles). However, in large but finite populations, the ef- additive model of variation in quantitative traits. Genetics 138: fect of increasing numbers of balanced polymorphisms 883–900. reaches a limit, beyond which adding more polymor- Charlesworth, B., and K. A. Hughes, 2000 The maintenance of genetic variation in life-history traits, pp. 369–392 in Evolutionary phisms does not further increase diversity (Navarro Genetics: From Molecules to Morphology, Vol. 1, edited by R. S. Singh and Barton 2002). Thus, it may be difficult to detect and C. B. Krimbas. Cambridge University Press, Cambridge, UK/ even widespread balancing selection through its effects London/New York. on neutral diversity. The most promising approach may Curtsinger, J. W., P. M. Service and T. Prout, 1994 Pleiotropy, reversal of dominance, and genetic polymorphism. Am. Nat. 144: be to ask whether variants that are actually under selec- 210–228. tion are often at intermediate frequency. This requires, De Luca, M., N. V. Roshina, G. L. Geiger-Thornsberry, R. F. Lyman, however, that the precise targets of selection be identi- E. G. Pasyukova et al., 2003 Dopa decarboxylase (Ddc) affects variation in Drosophila longevity. Nat. Genet. 34: 429–433. fied and that other processes that can raise allele fre- Dilda, C. L., and T. F. C. Mackay, 2002 The genetic architecture quencies (for example, hitchhiking) can be ruled out. of Drosophila sensory bristle number. Genetics 162: 1655–1674. Progress in the near future will be limited to model Falconer, D. S., and T. F. C. Mackay, 1996 Introduction to Quantita- tive Genetics, Ed. 4. Longman, Essex, UK. systems, such as D. melanogaster, in which plausible candi- Gantmacher, F. R., 1959 The Theory of Matrices, Vol. 1. Chelsea date loci and tools for fine-scale genetic analyses are Publishing, New York. available. Recent data (De Luca et al. 2003) suggest that Gavrilets, S., and A. Hastings, 1994 Maintenance of multilocus variability under strong stabilizing selection. J. Math. Biol. 32: the footprints of balancing selection may indeed be 287–302. observable for polygenic traits; but the generality of this Gillespie, J. H., 1974 Polymorphisms in random environments. observation remains to be determined. Theor. Popul. Biol. 4: 193–195. Gillespie, J. H., 1984 Pleiotropic overdominance and the mainte- We thank R. Bu¨rger, J. H. Gillespie, R. Haygood, T. F. C. Mackay, nance of genetic variation in polygenic characters. Genetics 107: S. V. Nuzhdin, M. Slatkin, and an anonymous reviewer for helpful com- 321–330. ments and discussion, and R. Haygood for implementing the multi- Gillespie, J. H., and H. A. Guess, 1978 The effects of environmental locus iterations. We thank the Erwin Schro¨dinger Institute for Mathe- autocorrelations on the progress of selection in a random envi- matical Physics at the University of Vienna for providing an excellent ronment. Am. Nat. 112: 897–910. research environment in which to complete this work, and N.H.B. thanks Gillespie, J., and C. Langley, 1976 Multilocus behavior in random the Center for Population Biology at University of California-Davis for environments. I. Random Levene models. Genetics 82: 123–137. its hospitality. This research was supported in part by National Science Gillespie, J. H., and M. Turelli, 1989 Genotype-environment interactions and the maintenance of polygenic variation. Genetics Foundation grant DEB-0089716 (M.T.) and plus grants GR3/11635 121: 129–138. from the Natural Environment Research Council and MMI09726 from Gillespie, J. H., and M. Turelli, 1990 Genotype-environment inter- the Engineering and Physical Sciences Research Council (N.H.B.). actions: a reply. Genetics 124: 447. Gimelfarb, A., 1989 Genotypic variation for a quantitative character maintained under stabilizing selection without mutations: epista- LITERATURE CITED sis. Genetics 123: 217–227. Gimelfarb, A., 1990 How much genetic variation can be maintained Andolfatto, P., F. Depaulis and A. Navarro, 2001 Inversion poly- by genotype-environment interactions? Genetics 124: 443–445. morphisms and nucleotide polymorphism: lessons from Droso- Gimelfarb, A., 1992 Pleiotropy and multilocus polymorphisms. Ge- phila. Genet. Res. 77: 1–8. netics 130: 223–227. Barton, N. 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Evolution POLYMORPHIC EQUILIBRIA 54: 13–20. Robertson, A., 1965 Synthesis, pp. 527–532 in Genetics Today, Vol. Here we establish that the eigenvalues of the stability 3, edited by S. J. Geerts. Pergamon Press, Oxford. matrix (10) are real and that necessary and sufficient Balancing Selection on Polygenes 1077 ␦ conditions for them all to be negative are (14) and (15), with ij as in (A2). By the logic used in (A4), we see that whereas (16) is necessary for stability. Our demonstra- C is positive definite if and only if V is. This demon- tion rests on the fact that the eigenvalues of a real strates that the stability conditions depend on only the ␣ symmetric matrix are real (Gantmacher 1959, Sect. vi and are independent of the mean effects, i, as well as IX.13) and on some elementary properties of deter- the equilibrium allele frequencies, pi. When is V positive minants and quadratic forms. First note that A can be definite? It is easy to derive a sufficient condition by written as considering the quadratic form associated with V. Note that for any vector x, A ϭϪSBC, (A1) n 2 n v Ϫ 1 T ϭ ϩ 2 i . with B a diagonal matrix whose ith diagonal element x V x ΂͚xi΃ ͚xi ΂ ΃ (A7) iϭ1 iϭ1 2 is 2piqi, the equilibrium heterozygosity at locus i [de- ϭ noted B diag(2piqi)], and C a symmetric, positive matrix This is obviously positive for all nonzero x if whose elements are Ͼ vi 1 for all i. (A8) Ϫ ϭ␣␣ ϩ␦ vi 1 ϭ cij i j ΄1 ij ΂ ΃΅ for i, j 1,...,n, (A2) Hence (A8) suffices for stability. The necessary condi- 2 tion (18) can be obtained by considering V as a variance- ␦ ␦ ϭ covariance matrix. To be positive definite, all correla- where ij denotes the Kronecker delta, with ii 1 and ␦ ϭ ϶ tions must be less than one. Thus, a necessary condition ij 0 for i j. Equation A1 implies that the eigenvalues of A are the eigenvalues of BC multiplied by ϪS. The for stability is ϩ ϩ Ͼ remaining argument has four steps. First we show that (vi 1)(vj 1) 4 (A9) the eigenvalues of BC are real, and then we show that ϶ they are all positive—implying that G ϫ E maintains a for all pairs with i j. stable polymorphic equilibrium—if and only if all of the Necessary and sufficient conditions follow from the fact that a real symmetric matrix is positive definite if eigenvalues of C are positive. Next, we reduce the prob- and only if all its “principal minors” (i.e., submatrices lem further by showing that we can set all of the ␣ ϭ i obtained by deleting rows and columns with identical 1 without loss of generality. The constraints on the v that i indices) have positive determinants (Gantmacher lead to stability then follow from properties of quadratic 1959, Sec. X.4). To present the conditions for stability forms. of the equilibria (9) concisely, assume that the v are Let B1/2 ϭ diag(√2pq) and BϪ1/2 ϭ diag(1/√2pq). Note i i i i i ordered from smallest, v , to largest, v . The necessary that BC ϭ BϪ1/2(B1/2CB1/2)B1/2. Hence, BC and B1/2 1 n and sufficient conditions are CB1/2 are “similar” matrices and have identical eigenval- m m m ues (Gantmacher 1959, Sec. III.6). Because B1/2CB1/2 Ϫ ϩ Ϫ Ͼ ͟(vi 1) 2͚ ͟(vj 1) 0, (A10) is real and symmetric, its eigenvalues are real. iϭ1 iϭ1 j϶i 1/2 1/2 Next we show that the eigenvalues of B CB (and for all m Յ n. Condition (A9), which corresponds to BC) are all positive if and only if the eigenvalues of C ϭ Ͼ Ն m 2 in (A10), implies that vi 1 for i 2. Consider are positive. Real symmetric matrices have all positive ϭ Ͻ (A10) with m n. Then, if v1 1, we must have eigenvalues if and only if they are positive definite 1 (Gantmacher 1959, Secs. X.4–5). Hence the eigen- v Ͼ 1 Ϫ . (A11) 1/2 1/2 1 ϩ n Ϫ values of B CB are all positive if and only if for all (1/2) ͚i ϭ2(1/(vi 1)) nonzero vectors x Thus, if there are many loci, or if some loci have vi near T 1/2 1/2 x B CB x Ͼ 0. (A3) 1, the lower bound on v1 will be not much below 1. Hence, for stable polygenic variation, the sufficient con- This is equivalent to Ͼ dition (A10), vi 1 for all i, is effectively also necessary. yTCy Ͼ 0 (A4) This qualitative conclusion is supported by our analysis of boundary equilibria [see (21) and appendix b]. for all nonzero y. Hence, stability of the polymorphic The special case in which only one locus is polymor- equilibrium is equivalent to determining when C is posi- phic deserves comment. In this case, the stability condi- tive definite. tion reduces to The problem can be simplified further by noting that ϾϪ v1 1. (A12) ϭ C DVD, (A5) This trivially generalizes Wright’s (1935) result by ϭ ␣ ␣ ␣ where D diag( 1, 2,..., n) and V is the positive, showing that even underdominant selection can be bal- symmetric matrix with elements anced by stabilizing selection to retain one polymorphic locus if the heterozygote produces a near-optimal phe- Ϫ ϭ ϩ␦ vi 1 notype in a genetic background in which all other loci vij 1 ij ΂ ΃, (A6) 2 are fixed. 1078 M. Turelli and N. H. Barton

APPENDIX B: STABILITY CONDITIONS FOR Conditions (19b) require BOUNDARY EQUILIBRIA ␣ ˆ Ϫ Ͻ ⌬ ␣ Ϫ ϽϪ ⌬ ϭ I(2pIvI 1) 2 and I(2qÎvI 1) 2 . Let A (aij) denote the stability matrix corresponding ϭ ʦ ⍀ ϭ ʦ (B4) to an equilibrium with pi 0 for i 0, pi 1 for i ⍀ Ͻ Ͻ ʦ ⍀ 1, and 0 pi 1 for i p. As shown by (20), the Overall, the constraints on the one polymorphic locus Ͻ eigenvalues governing the stability of the loci fixed at with vi 1 are very restrictive. ␭ ϭ ʦ ⍀ ʜ ⍀ 0 and 1 are simply i aii for i 0 1 (20b and 20c). Moreover, the eigenvalues governing the polymor- APPENDIX C: FINDING ALTERNATIVE phic loci are generated by a matrix whose elements are STABLE EQUILIBRIA given by (10). The stability conditions for this subsystem ⍀ ˆ ϭ are given by (14) and (15) if p has at least two elements We restrict attention to the special case pi 0.5 and [or (A14) if there is just one polymorphic locus] and ␪ϭ0 considered in our first numerical example con- depend only on the vi. What remains is to find the cerning pleiotropic balancing selection. Because of the conditions for stability of the fixed equilibria in an ex- symmetry, we can restrict attention to equilibria with plicit form. ⌬Ͼ0, as the complementary equilibria with ⌬Ͻ0 can For definiteness, suppose that the mean lies above be found by simply reversing the frequencies of Bi and ⌬Ͼ ␣ Ͼ ␦ Ͼ the optimum ( 0) and that all i 0 (hence i bi at each locus. We seek an exhaustive list of multilocus 0); the argument is similar for the other cases. With equilibria that satisfy our stability and feasibility con- ⌬Ͼ0, directional selection on the trait is tending to straints. The key idea is to recognize that these alterna- reduce the pi. First, suppose that the polymorphic loci tive equilibria fall into classes that are determined by all satisfy v Ͼ 1. Then, from (23), ⌬ Ͼ 0. For the the relationship of 2⌬ to the intervals defined by the i f Ϯ␣ Ϫ polymorphic equilibria to be feasible, (19a) implies that sequence of values for i(vi 1). For any assignment ␣ Ϯ ␣ Ϫ we must have of the i and vi, we first order the sequence | i(vi 1)| and then find the possible equilibria that fall into ⌬ 2 f ␣ (2pˆ v Ϫ 1) Ͼ 2⌬ϭ (B1a) each of these intervals, including the regions below mini i i i ϩ Ϫ ␣ Ϫ ␣ Ϫ 1 2C { | i(vi 1)|} and above maxi{| i(vi 1)|}. The strategy is to start with a trial value of ␹ϭ2⌬ in and one of these intervals, then to determine for this value ⌬ ␣ Ϫ ϾϪ ⌬ϭ Ϫ 2 f ʦ ⍀ the sets of loci that can be fixed for 0, fixed for 1, or i(2qîvi 1) 2 for i p , 1 ϩ 2C polymorphic, and then from these to determine which ⌬ (B1b) values of in the interval being considered can be realized. The crucial observation is that according to ⌬ Ͼ with f and C as in (23). Note that even with vi 1, our stability and feasibility conditions, the qualitative (B1a) implies that polymorphism may not be feasible equilibrium state of each locus depends on only the ⌬ Ͻ for any f if pî is too small (e.g., pî 1/2vi). In contrast, value of ␣ (v Ϫ 1) relative to Ϯ⌬. Because of this, we ϭ Ͼ i i if pî 1/2, then vi 1 ensures that polymorphism is need consider only one trial value of ␹ in each interval. ⌬ feasible if f is sufficiently small. Note that conditions First, consider equilibria where all polymorphic loci (B1) are more easily satisfied with large C, correspond- Ͼ ␹ϭ ⌬ have vi 1. A trial value of the threshold 2 is ing to many polymorphic loci and/or relatively weak chosen. If ␹Ͻ0, then the complementary case is consid- Ͼ balancing selection (i.e., vi slightly 1). For the fixed ered, with ␹Ͼ0; loci fixed for 0 and 1 are then reversed. loci to be stable, conditions (21) imply that For fixed ␹, all of the loci are sorted into three classes ␣ Ϫ ␣ (2pˆ v Ϫ 1) Ͻ 2⌬ for i ʦ ⍀ , (B2a) according to their value of i(vi 1). According to i i i 0 Ͼ ␣ Ϫ Ͼ␹ (B2), those loci with vi 1 and i(vi 1) must be ␹Ͼ␣ Ϫ ϾϪ␹ and polymorphic, and those with i(vi 1) must Ϫ␹ Ͼ ␣ Ϫ ␣ Ϫ ϽϪ ⌬ ʦ ⍀ fix at 0. Finally, those with i(vi 1) may fix at i(2qîvi 1) 2 for i 1 . (B2b) 0 or 1. Thus, the task reduces to considering all of the ϭ ϭ Thus, for pî 1/2, loci with pi 1 can be stable only equilibria with the loci in this final class fixed for either ␣ Ϫ ϽϪ⌬ Ͻ if i(vi 1) 2 , which requires vi 1, but loci 0 or 1 and determining which of these configurations ϭ Ͼ ␣ Ϫ ⌬ may fix for pi 0 even if vi 1, provided that i(vi produces values of in the interval being considered. 1) Ͻ 2⌬. All that do produce them are feasible stable equilibria. Ͻ ʦ ␹ Now, consider an equilibrium with vI 1 for one I By trying values of in each interval, all possible stable ⍀ ⌬Ͼ p. As before, we assume that 0. From stability con- equilibria can be found. Ͼ ʦ ⍀ ϶ ␹ dition (A9), we know that vi 1 for i p if i I. Finally, for each , we consider possible stable equilib- Ͻ Then, from (A10), 1 ϩ 2C Ͻ 0, with C as in (23c). ria at which one of the polymorphic loci has vi 1. Let Condition (A11) implies that I denote such a locus. For it to be feasibly polymorphic, ␣ Ϫ ϽϪ␹ it must satisfy I(vI 1) (B2b), and it must also Ͼ Ϫ 2 Ͼ vI 1 ΂ ΃ 0. (B3) satisfy the stability criterion (B3). This condition can 1 ϩ 2͚ ʦ ⍀ Ϫ (1/(v Ϫ 1)) ⌬ ␪ Ͼ j { p I } j be written as 2 / eff 0. Jointly, these conditions are Balancing Selection on Polygenes 1079 very restrictive. The algorithm is to find all loci that assign them individually as polymorphic, and then carry satisfy out the procedure described above with all of the re- Ͼ ␣ maining polymorphic loci satisfying vi 1. Configurations Ϫ 2 I Ͻ␣ Ϫ ϽϪ␹ ⌬ I(vI 1) , (C1) of monomorphic loci that lead to values of 2 in the ϩ ͚ ʦ ⍀ Ϫ Ϫ 1 2 j { p I } (1/(vj 1)) target interval are accepted as stable equilibria.